[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
p q : NonUnitalStarSubalgebra R A
h : (fun {s} => s.carrier) = fun {s} => s.carrier
⊢ p = q
[PROOFSTEP]
cases p
[GOAL]
case mk
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
q : NonUnitalStarSubalgebra R A
toNonUnitalSubalgebra✝ : NonUnitalSubalgebra R A
star_mem'✝ : ∀ {a : A}, a ∈ toNonUnitalSubalgebra✝.carrier → star a ∈ toNonUnitalSubalgebra✝.carrier
h : (fun {s} => s.carrier) = fun {s} => s.carrier
⊢ { toNonUnitalSubalgebra := toNonUnitalSubalgebra✝, star_mem' := star_mem'✝ } = q
[PROOFSTEP]
cases q
[GOAL]
case mk.mk
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
toNonUnitalSubalgebra✝¹ : NonUnitalSubalgebra R A
star_mem'✝¹ : ∀ {a : A}, a ∈ toNonUnitalSubalgebra✝¹.carrier → star a ∈ toNonUnitalSubalgebra✝¹.carrier
toNonUnitalSubalgebra✝ : NonUnitalSubalgebra R A
star_mem'✝ : ∀ {a : A}, a ∈ toNonUnitalSubalgebra✝.carrier → star a ∈ toNonUnitalSubalgebra✝.carrier
h : (fun {s} => s.carrier) = fun {s} => s.carrier
⊢ { toNonUnitalSubalgebra := toNonUnitalSubalgebra✝¹, star_mem' := star_mem'✝¹ } =
    { toNonUnitalSubalgebra := toNonUnitalSubalgebra✝, star_mem' := star_mem'✝ }
[PROOFSTEP]
congr
[GOAL]
case mk.mk.e_toNonUnitalSubalgebra
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
toNonUnitalSubalgebra✝¹ : NonUnitalSubalgebra R A
star_mem'✝¹ : ∀ {a : A}, a ∈ toNonUnitalSubalgebra✝¹.carrier → star a ∈ toNonUnitalSubalgebra✝¹.carrier
toNonUnitalSubalgebra✝ : NonUnitalSubalgebra R A
star_mem'✝ : ∀ {a : A}, a ∈ toNonUnitalSubalgebra✝.carrier → star a ∈ toNonUnitalSubalgebra✝.carrier
h : (fun {s} => s.carrier) = fun {s} => s.carrier
⊢ toNonUnitalSubalgebra✝¹ = toNonUnitalSubalgebra✝
[PROOFSTEP]
exact SetLike.coe_injective h
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
h : S.toNonUnitalSubalgebra = T.toNonUnitalSubalgebra
x : A
⊢ x ∈ S ↔ x ∈ T
[PROOFSTEP]
rw [← mem_toNonUnitalSubalgebra, ← mem_toNonUnitalSubalgebra, h]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
s : Set A
hs : s = ↑S
src✝ : NonUnitalSubalgebra R A := NonUnitalSubalgebra.copy S.toNonUnitalSubalgebra s hs
x : A
hx : x ∈ s
⊢ star x ∈
    { toNonUnitalSubsemiring := src✝.toNonUnitalSubsemiring,
              smul_mem' :=
                (_ :
                  ∀ (c : R) {x : A},
                    x ∈ src✝.carrier →
                      c • x ∈ src✝.carrier) }.toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier
[PROOFSTEP]
show star x ∈ s
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
s : Set A
hs : s = ↑S
src✝ : NonUnitalSubalgebra R A := NonUnitalSubalgebra.copy S.toNonUnitalSubalgebra s hs
x : A
hx : x ∈ s
⊢ star x ∈ s
[PROOFSTEP]
rw [hs] at hx ⊢
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
s : Set A
hs : s = ↑S
src✝ : NonUnitalSubalgebra R A := NonUnitalSubalgebra.copy S.toNonUnitalSubalgebra s hs
x : A
hx : x ∈ ↑S
⊢ star x ∈ ↑S
[PROOFSTEP]
exact S.star_mem' hx
[GOAL]
F : Type v'
R' : Type u'
R✝ : Type u
A✝ : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R✝
inst✝¹³ : NonUnitalNonAssocSemiring A✝
inst✝¹² : Module R✝ A✝
inst✝¹¹ : Star A✝
inst✝¹⁰ : NonUnitalNonAssocSemiring B
inst✝⁹ : Module R✝ B
inst✝⁸ : Star B
inst✝⁷ : NonUnitalNonAssocSemiring C
inst✝⁶ : Module R✝ C
inst✝⁵ : Star C
inst✝⁴ : NonUnitalStarAlgHomClass F R✝ A✝ B
S✝ : NonUnitalStarSubalgebra R✝ A✝
R : Type u
A : Type v
inst✝³ : CommRing R
inst✝² : NonUnitalRing A
inst✝¹ : Module R A
inst✝ : Star A
S T : NonUnitalStarSubalgebra R A
h : toNonUnitalSubring S = toNonUnitalSubring T
x : A
⊢ x ∈ S ↔ x ∈ T
[PROOFSTEP]
rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
S✝ S T : NonUnitalStarSubalgebra R A
h : (fun S => S.toNonUnitalSubalgebra) S = (fun S => S.toNonUnitalSubalgebra) T
⊢ ∀ (x : A), x ∈ S ↔ x ∈ T
[PROOFSTEP]
apply SetLike.ext_iff.1 h
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
S✝ : NonUnitalStarSubalgebra R A
f : F
S : NonUnitalStarSubalgebra R A
⊢ ∀ {a : B},
    a ∈
        (NonUnitalSubalgebra.map (↑f)
                  S.toNonUnitalSubalgebra).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier →
      star a ∈
        (NonUnitalSubalgebra.map (↑f)
                  S.toNonUnitalSubalgebra).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier
[PROOFSTEP]
rintro _ ⟨a, ha, rfl⟩
[GOAL]
case intro.intro
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
S✝ : NonUnitalStarSubalgebra R A
f : F
S : NonUnitalStarSubalgebra R A
a : A
ha : a ∈ ↑S.toNonUnitalSubsemiring
⊢ star (↑↑↑f a) ∈
    (NonUnitalSubalgebra.map (↑f)
              S.toNonUnitalSubalgebra).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier
[PROOFSTEP]
exact ⟨star a, star_mem (s := S) ha, map_star f a⟩
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
φ : F
⊢ ∀ {a : B},
    a ∈ (NonUnitalAlgHom.range ↑φ).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier →
      star a ∈ (NonUnitalAlgHom.range ↑φ).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier
[PROOFSTEP]
rintro _ ⟨a, rfl⟩
[GOAL]
case intro
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
φ : F
a : A
⊢ star (↑↑↑φ a) ∈ (NonUnitalAlgHom.range ↑φ).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier
[PROOFSTEP]
exact ⟨star a, map_star φ a⟩
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
φ : F
⊢ ↑(NonUnitalStarAlgHom.range φ) = Set.range ↑φ
[PROOFSTEP]
ext
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
φ : F
x✝ : B
⊢ x✝ ∈ ↑(NonUnitalStarAlgHom.range φ) ↔ x✝ ∈ Set.range ↑φ
[PROOFSTEP]
rw [SetLike.mem_coe, mem_range]
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
φ : F
x✝ : B
⊢ (∃ x, ↑φ x = x✝) ↔ x✝ ∈ Set.range ↑φ
[PROOFSTEP]
rfl
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : Star A
inst✝⁶ : NonUnitalNonAssocSemiring B
inst✝⁵ : Module R B
inst✝⁴ : Star B
inst✝³ : NonUnitalNonAssocSemiring C
inst✝² : Module R C
inst✝¹ : Star C
inst✝ : NonUnitalStarAlgHomClass F R A B
ϕ ψ : F
x : A
hx : ↑ϕ x = ↑ψ x
⊢ star x ∈ (NonUnitalAlgHom.equalizer ϕ ψ).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier
[PROOFSTEP]
simp [map_star, hx]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Module R A
inst✝⁶ : Star A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : Module R B
inst✝³ : Star B
inst✝² : NonUnitalSemiring C
inst✝¹ : Module R C
inst✝ : Star C
hF : NonUnitalStarAlgHomClass F R A B
g : B → A
f : F
h : Function.LeftInverse g ↑f
src✝ : A →⋆ₙₐ[R] { x // x ∈ NonUnitalStarAlgHom.range f } := NonUnitalStarAlgHom.rangeRestrict f
x : { x // x ∈ NonUnitalStarAlgHom.range f }
x' : A
hx' : ↑f x' = ↑x
⊢ ↑f (g ↑x) = ↑x
[PROOFSTEP]
rw [← hx', h x']
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
S : NonUnitalSubalgebra R A
x y : A
hx : x ∈ star S.carrier
hy : y ∈ star S.carrier
⊢ x + y ∈ star S.carrier
[PROOFSTEP]
simpa only [Set.mem_star, NonUnitalSubalgebra.mem_carrier] using (star_add x y).symm ▸ add_mem hx hy
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
S : NonUnitalSubalgebra R A
x y : A
hx :
  x ∈
    {
          toAddSubsemigroup :=
            { carrier := star S.carrier,
              add_mem' := (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
          zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier
hy :
  y ∈
    {
          toAddSubsemigroup :=
            { carrier := star S.carrier,
              add_mem' := (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
          zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier
⊢ x * y ∈
    {
          toAddSubsemigroup :=
            { carrier := star S.carrier,
              add_mem' := (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
          zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier
[PROOFSTEP]
simpa only [Set.mem_star, NonUnitalSubalgebra.mem_carrier] using (star_mul x y).symm ▸ mul_mem hy hx
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
S : NonUnitalSubalgebra R A
r : R
x : A
hx :
  x ∈
    {
            toAddSubmonoid :=
              {
                toAddSubsemigroup :=
                  { carrier := star S.carrier,
                    add_mem' := (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) },
            mul_mem' :=
              (_ :
                ∀ (x y : A),
                  x ∈
                      {
                            toAddSubsemigroup :=
                              { carrier := star S.carrier,
                                add_mem' :=
                                  (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                            zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier →
                    y ∈
                        {
                              toAddSubsemigroup :=
                                { carrier := star S.carrier,
                                  add_mem' :=
                                    (_ :
                                      ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                              zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier →
                      x * y ∈
                        {
                              toAddSubsemigroup :=
                                { carrier := star S.carrier,
                                  add_mem' :=
                                    (_ :
                                      ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                              zero_mem' :=
                                (_ :
                                  star 0 ∈
                                    S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier
⊢ r • x ∈
    {
            toAddSubmonoid :=
              {
                toAddSubsemigroup :=
                  { carrier := star S.carrier,
                    add_mem' := (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) },
            mul_mem' :=
              (_ :
                ∀ (x y : A),
                  x ∈
                      {
                            toAddSubsemigroup :=
                              { carrier := star S.carrier,
                                add_mem' :=
                                  (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                            zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier →
                    y ∈
                        {
                              toAddSubsemigroup :=
                                { carrier := star S.carrier,
                                  add_mem' :=
                                    (_ :
                                      ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                              zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier →
                      x * y ∈
                        {
                              toAddSubsemigroup :=
                                { carrier := star S.carrier,
                                  add_mem' :=
                                    (_ :
                                      ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                              zero_mem' :=
                                (_ :
                                  star 0 ∈
                                    S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier
[PROOFSTEP]
simpa only [Set.mem_star, NonUnitalSubalgebra.mem_carrier] using
  (star_smul r x).symm ▸ SMulMemClass.smul_mem (star r) hx
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
S : NonUnitalSubalgebra R A
x : A
⊢ star x ∈ star S ↔ x ∈ S
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
s : Set A
this : ∀ (t : Set A), NonUnitalAlgebra.adjoin R (star t) ≤ star (NonUnitalAlgebra.adjoin R t)
⊢ star (NonUnitalAlgebra.adjoin R s) ≤ NonUnitalAlgebra.adjoin R (star s)
[PROOFSTEP]
simpa only [star_star] using NonUnitalSubalgebra.star_mono (this (star s))
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
S : NonUnitalSubalgebra R A
a : A
ha : a ∈ S ⊔ star S
⊢ star a ∈ S ⊔ star S
[PROOFSTEP]
simp only [← mem_star_iff _ a, ← (@NonUnitalAlgebra.gi R A _ _ _ _ _).l_sup_u _ _] at *
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
S : NonUnitalSubalgebra R A
a : A
ha : a ∈ NonUnitalAlgebra.adjoin R (↑S ⊔ ↑(star S))
⊢ a ∈ star (NonUnitalAlgebra.adjoin R (↑S ⊔ ↑(star S)))
[PROOFSTEP]
convert ha using 2
[GOAL]
case h.e'_5
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
S : NonUnitalSubalgebra R A
a : A
ha : a ∈ NonUnitalAlgebra.adjoin R (↑S ⊔ ↑(star S))
⊢ star (NonUnitalAlgebra.adjoin R (↑S ⊔ ↑(star S))) = NonUnitalAlgebra.adjoin R (↑S ⊔ ↑(star S))
[PROOFSTEP]
simp only [Set.sup_eq_union, star_adjoin_comm, Set.union_star, coe_star, star_star, Set.union_comm]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
s : Set A
a✝ : A
x✝ : a✝ ∈ (NonUnitalAlgebra.adjoin R (s ∪ star s)).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier
⊢ star a✝ ∈ (NonUnitalAlgebra.adjoin R (s ∪ star s)).toNonUnitalSubsemiring.toAddSubmonoid.toAddSubsemigroup.carrier
[PROOFSTEP]
rwa [NonUnitalSubalgebra.mem_carrier, ← NonUnitalSubalgebra.mem_star_iff, NonUnitalSubalgebra.star_adjoin_comm,
  Set.union_star, star_star, Set.union_comm]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
⊢ GaloisConnection (adjoin R) SetLike.coe
[PROOFSTEP]
intro s S
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
s : Set A
S : NonUnitalStarSubalgebra R A
⊢ adjoin R s ≤ S ↔ s ≤ ↑S
[PROOFSTEP]
rw [← toNonUnitalSubalgebra_le_iff, adjoin_toNonUnitalSubalgebra, NonUnitalAlgebra.adjoin_le_iff,
  coe_toNonUnitalSubalgebra]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
s : Set A
S : NonUnitalStarSubalgebra R A
⊢ s ∪ star s ⊆ ↑S ↔ s ≤ ↑S
[PROOFSTEP]
exact
  ⟨fun h => (Set.subset_union_left s _).trans h, fun h =>
    Set.union_subset h fun x hx => star_star x ▸ star_mem (show star x ∈ S from h hx)⟩
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
⊢ ⊤.toNonUnitalSubalgebra = ⊤
[PROOFSTEP]
ext
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
x✝ : A
⊢ x✝ ∈ ⊤.toNonUnitalSubalgebra ↔ x✝ ∈ ⊤
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ ∀ {x : A}, x ∈ S → x ∈ S ⊔ T
[PROOFSTEP]
rw [← SetLike.le_def]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ S ≤ S ⊔ T
[PROOFSTEP]
exact le_sup_left
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ ∀ {x : A}, x ∈ T → x ∈ S ⊔ T
[PROOFSTEP]
rw [← SetLike.le_def]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ T ≤ S ⊔ T
[PROOFSTEP]
exact le_sup_right
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : Set (NonUnitalStarSubalgebra R A)
x : A
⊢ x ∈ sInf S ↔ ∀ (p : NonUnitalStarSubalgebra R A), p ∈ S → x ∈ p
[PROOFSTEP]
simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : Set (NonUnitalStarSubalgebra R A)
⊢ ↑(sInf S).toNonUnitalSubalgebra = ↑(sInf (toNonUnitalSubalgebra '' S))
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
ι : Sort u_1
S : ι → NonUnitalStarSubalgebra R A
⊢ ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)
[PROOFSTEP]
simp [iInf]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
ι : Sort u_1
S : ι → NonUnitalStarSubalgebra R A
x : A
⊢ x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i
[PROOFSTEP]
simp only [iInf, mem_sInf, Set.forall_range_iff]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
ι : Sort u_1
S : ι → NonUnitalStarSubalgebra R A
⊢ ↑(⨅ (i : ι), S i).toNonUnitalSubalgebra = ↑(⨅ (i : ι), (S i).toNonUnitalSubalgebra)
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
x : A
⊢ x ∈ NonUnitalAlgebra.adjoin R (∅ ∪ star ∅) ↔ x = 0
[PROOFSTEP]
rw [Set.star_empty, Set.union_empty, NonUnitalAlgebra.adjoin_empty, NonUnitalAlgebra.mem_bot]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
⊢ ⊥.toNonUnitalSubalgebra = ⊥
[PROOFSTEP]
ext x
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
x : A
⊢ x ∈ ⊥.toNonUnitalSubalgebra ↔ x ∈ ⊥
[PROOFSTEP]
simp only [mem_bot, NonUnitalStarSubalgebra.mem_toNonUnitalSubalgebra, NonUnitalAlgebra.mem_bot]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
⊢ ↑⊥ = {0}
[PROOFSTEP]
simp only [Set.ext_iff, NonUnitalStarAlgebra.mem_bot, SetLike.mem_coe, Set.mem_singleton_iff, iff_self_iff,
  forall_const]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
h : S = ⊤
x : A
⊢ x ∈ S
[PROOFSTEP]
rw [h]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
h : S = ⊤
x : A
⊢ x ∈ ⊤
[PROOFSTEP]
exact mem_top
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
h : ∀ (x : A), x ∈ S
⊢ S = ⊤
[PROOFSTEP]
ext x
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
h : ∀ (x : A), x ∈ S
x : A
⊢ x ∈ S ↔ x ∈ ⊤
[PROOFSTEP]
exact ⟨fun _ => mem_top, fun _ => h x⟩
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
f : F
⊢ ↑(map f ⊥) = ↑⊥
[PROOFSTEP]
simp [NonUnitalAlgebra.coe_bot, NonUnitalStarSubalgebra.coe_map]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B✝ : Type w
C✝ : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B✝
inst✝⁵ : StarRing B✝
inst✝⁴ : Module R B✝
inst✝³ : IsScalarTower R B✝ B✝
inst✝² : SMulCommClass R B✝ B✝
inst✝¹ : StarModule R B✝
hF : NonUnitalStarAlgHomClass F R A B✝
S : NonUnitalStarSubalgebra R A
inst✝ : Subsingleton A
B C : NonUnitalStarSubalgebra R A
x : A
⊢ x ∈ B ↔ x ∈ C
[PROOFSTEP]
simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
S₁ : NonUnitalStarSubalgebra R B
⊢ prod ⊤ ⊤ = ⊤
[PROOFSTEP]
ext
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
S₁ : NonUnitalStarSubalgebra R B
x✝ : A × B
⊢ x✝ ∈ prod ⊤ ⊤ ↔ x✝ ∈ ⊤
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
⊢ { x // x ∈ T } →⋆ₙₐ[R] B
[PROOFSTEP]
subst hT
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ { x // x ∈ iSup K } →⋆ₙₐ[R] B
[PROOFSTEP]
exact
  { toFun :=
      Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)
        (fun i j x hxi hxj => by
          let ⟨k, hik, hjk⟩ := dir i j
          simp only
          rw [hf i k hik, hf j k hjk]
          rfl)
        (↑(iSup K)) (by rw [coe_iSup_of_directed dir])
    map_zero' :=
      by
      dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply, inclusion_mk, Eq.ndrec, id_eq,
        eq_mpr_eq_cast]
      exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)
    map_mul' :=
      by
      dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply, inclusion_mk, Eq.ndrec, id_eq,
        eq_mpr_eq_cast, ZeroMemClass.coe_zero, AddSubmonoid.mk_add_mk, Set.inclusion_mk]
      apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))
      on_goal 3 => rw [coe_iSup_of_directed dir]
      all_goals simp
    map_add' :=
      by
      dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply, inclusion_mk, Eq.ndrec, id_eq,
        eq_mpr_eq_cast]
      apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))
      on_goal 3 => rw [coe_iSup_of_directed dir]
      all_goals simp
    map_smul' := fun r =>
      by
      dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply, inclusion_mk, Eq.ndrec, id_eq,
        eq_mpr_eq_cast]
      apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x) (fun _ _ => rfl)
      on_goal 2 => rw [coe_iSup_of_directed dir]
      all_goals simp
    map_star' :=
      by
      dsimp only [SetLike.coe_sort_coe, NonUnitalStarAlgHom.comp_apply, inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast,
        ZeroMemClass.coe_zero, AddSubmonoid.mk_add_mk, Set.inclusion_mk, MulMemClass.mk_mul_mk,
        NonUnitalAlgHom.toDistribMulActionHom_eq_coe, DistribMulActionHom.toFun_eq_coe,
        NonUnitalAlgHom.coe_to_distribMulActionHom, NonUnitalAlgHom.coe_mk]
      apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => star x) (fun _ _ => rfl)
      on_goal 2 => rw [coe_iSup_of_directed dir]
      all_goals simp [map_star] }
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i j : ι
x : A
hxi : x ∈ (fun i => ↑(K i)) i
hxj : x ∈ (fun i => ↑(K i)) j
⊢ (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj }
[PROOFSTEP]
let ⟨k, hik, hjk⟩ := dir i j
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i j : ι
x : A
hxi : x ∈ (fun i => ↑(K i)) i
hxj : x ∈ (fun i => ↑(K i)) j
k : ι
hik : (fun x x_1 => x ≤ x_1) (K i) (K k)
hjk : (fun x x_1 => x ≤ x_1) (K j) (K k)
⊢ (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj }
[PROOFSTEP]
simp only
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i j : ι
x : A
hxi : x ∈ (fun i => ↑(K i)) i
hxj : x ∈ (fun i => ↑(K i)) j
k : ι
hik : (fun x x_1 => x ≤ x_1) (K i) (K k)
hjk : (fun x x_1 => x ≤ x_1) (K j) (K k)
⊢ ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj }
[PROOFSTEP]
rw [hf i k hik, hf j k hjk]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i j : ι
x : A
hxi : x ∈ (fun i => ↑(K i)) i
hxj : x ∈ (fun i => ↑(K i)) j
k : ι
hik : (fun x x_1 => x ≤ x_1) (K i) (K k)
hjk : (fun x x_1 => x ≤ x_1) (K j) (K k)
⊢ ↑(NonUnitalStarAlgHom.comp (f k) (inclusion hik)) { val := x, property := hxi } =
    ↑(NonUnitalStarAlgHom.comp (f k) (inclusion hjk)) { val := x, property := hxj }
[PROOFSTEP]
rfl
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
rw [coe_iSup_of_directed dir]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ∀ (x : { x // x ∈ iSup K }),
    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
        (_ :
          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
              (fun i x => ↑(f i) x) j { val := x, property := hxj })
        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
      r •
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
          (_ :
            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                (fun i x => ↑(f i) x) j { val := x, property := hxj })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x
[PROOFSTEP]
dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply, inclusion_mk, Eq.ndrec, id_eq,
  eq_mpr_eq_cast]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ∀ (x : { x // x ∈ iSup K }),
    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
        (_ :
          ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
            ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
      r •
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
          (_ :
            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
              ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x
[PROOFSTEP]
apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x) (fun _ _ => rfl)
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ∀ (i : ι) (x : ↑↑(K i)), ↑(f i) (r • x) = r • ↑(f i) x
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
on_goal 2 => rw [coe_iSup_of_directed dir]
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ∀ (i : ι) (x : ↑↑(K i)), ↑(f i) (r • x) = r • ↑(f i) x
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
on_goal 2 => rw [coe_iSup_of_directed dir]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
rw [coe_iSup_of_directed dir]
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ∀ (i : ι) (x : ↑↑(K i)), ↑(f i) (r • x) = r • ↑(f i) x
[PROOFSTEP]
all_goals simp
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
r : R
⊢ ∀ (i : ι) (x : ↑↑(K i)), ↑(f i) (r • x) = r • ↑(f i) x
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ MulActionHom.toFun
      {
        toFun :=
          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
            (_ :
              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
        map_smul' :=
          (_ :
            ∀ (r : R) (x : { x // x ∈ iSup K }),
              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                  (_ :
                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                r •
                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                    (_ :
                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
      0 =
    0
[PROOFSTEP]
dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply, inclusion_mk, Eq.ndrec, id_eq,
  eq_mpr_eq_cast]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
      (_ :
        ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
          ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) 0 =
    0
[PROOFSTEP]
exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι), ↑(f i) ((fun i => 0) i) = 0
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (x y : { x // x ∈ iSup K }),
    MulActionHom.toFun
        {
          toFun :=
            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
              (_ :
                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
          map_smul' :=
            (_ :
              ∀ (r : R) (x : { x // x ∈ iSup K }),
                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                    (_ :
                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                  r •
                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                      (_ :
                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
        (x + y) =
      MulActionHom.toFun
          {
            toFun :=
              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                (_ :
                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
            map_smul' :=
              (_ :
                ∀ (r : R) (x : { x // x ∈ iSup K }),
                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                      (_ :
                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                    r •
                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                        (_ :
                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
          x +
        MulActionHom.toFun
          {
            toFun :=
              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                (_ :
                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
            map_smul' :=
              (_ :
                ∀ (r : R) (x : { x // x ∈ iSup K }),
                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                      (_ :
                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                    r •
                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                        (_ :
                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
          y
[PROOFSTEP]
dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply, inclusion_mk, Eq.ndrec, id_eq,
  eq_mpr_eq_cast]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (x y : { x // x ∈ iSup K }),
    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
        (_ :
          ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
            ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (x + y) =
      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
          (_ :
            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
              ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x +
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
          (_ :
            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
              ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) y
[PROOFSTEP]
apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))
[GOAL]
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), ↑(f i) (x + y) = ↑(f i) x + ↑(f i) y
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
on_goal 3 => rw [coe_iSup_of_directed dir]
[GOAL]
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), ↑(f i) (x + y) = ↑(f i) x + ↑(f i) y
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
on_goal 3 => rw [coe_iSup_of_directed dir]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
rw [coe_iSup_of_directed dir]
[GOAL]
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), ↑(f i) (x + y) = ↑(f i) x + ↑(f i) y
[PROOFSTEP]
all_goals simp
[GOAL]
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
[PROOFSTEP]
simp
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), ↑(f i) (x + y) = ↑(f i) x + ↑(f i) y
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (x y : { x // x ∈ iSup K }),
    MulActionHom.toFun
        {
            toMulActionHom :=
              {
                toFun :=
                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                    (_ :
                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                map_smul' :=
                  (_ :
                    ∀ (r : R) (x : { x // x ∈ iSup K }),
                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                          (_ :
                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                        r •
                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                            (_ :
                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
            map_zero' :=
              (_ :
                MulActionHom.toFun
                    {
                      toFun :=
                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                          (_ :
                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                      map_smul' :=
                        (_ :
                          ∀ (r : R) (x : { x // x ∈ iSup K }),
                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                (_ :
                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                              r •
                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                  (_ :
                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                    0 =
                  0),
            map_add' :=
              (_ :
                ∀ (x y : { x // x ∈ iSup K }),
                  MulActionHom.toFun
                      {
                        toFun :=
                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                            (_ :
                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                        map_smul' :=
                          (_ :
                            ∀ (r : R) (x : { x // x ∈ iSup K }),
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                  (_ :
                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                r •
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                      (x + y) =
                    MulActionHom.toFun
                        {
                          toFun :=
                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                              (_ :
                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                          map_smul' :=
                            (_ :
                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                  r •
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                        x +
                      MulActionHom.toFun
                        {
                          toFun :=
                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                              (_ :
                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                          map_smul' :=
                            (_ :
                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                  r •
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                        y) }.toMulActionHom
        (x * y) =
      MulActionHom.toFun
          {
              toMulActionHom :=
                {
                  toFun :=
                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                      (_ :
                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                  map_smul' :=
                    (_ :
                      ∀ (r : R) (x : { x // x ∈ iSup K }),
                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                            (_ :
                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                          r •
                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                              (_ :
                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
              map_zero' :=
                (_ :
                  MulActionHom.toFun
                      {
                        toFun :=
                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                            (_ :
                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                        map_smul' :=
                          (_ :
                            ∀ (r : R) (x : { x // x ∈ iSup K }),
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                  (_ :
                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                r •
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                      0 =
                    0),
              map_add' :=
                (_ :
                  ∀ (x y : { x // x ∈ iSup K }),
                    MulActionHom.toFun
                        {
                          toFun :=
                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                              (_ :
                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                          map_smul' :=
                            (_ :
                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                  r •
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                        (x + y) =
                      MulActionHom.toFun
                          {
                            toFun :=
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                (_ :
                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                            map_smul' :=
                              (_ :
                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                    r •
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                          x +
                        MulActionHom.toFun
                          {
                            toFun :=
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                (_ :
                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                            map_smul' :=
                              (_ :
                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                    r •
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                          y) }.toMulActionHom
          x *
        MulActionHom.toFun
          {
              toMulActionHom :=
                {
                  toFun :=
                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                      (_ :
                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                  map_smul' :=
                    (_ :
                      ∀ (r : R) (x : { x // x ∈ iSup K }),
                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                            (_ :
                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                          r •
                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                              (_ :
                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
              map_zero' :=
                (_ :
                  MulActionHom.toFun
                      {
                        toFun :=
                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                            (_ :
                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                        map_smul' :=
                          (_ :
                            ∀ (r : R) (x : { x // x ∈ iSup K }),
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                  (_ :
                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                r •
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                      0 =
                    0),
              map_add' :=
                (_ :
                  ∀ (x y : { x // x ∈ iSup K }),
                    MulActionHom.toFun
                        {
                          toFun :=
                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                              (_ :
                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                          map_smul' :=
                            (_ :
                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                  r •
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                        (x + y) =
                      MulActionHom.toFun
                          {
                            toFun :=
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                (_ :
                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                            map_smul' :=
                              (_ :
                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                    r •
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                          x +
                        MulActionHom.toFun
                          {
                            toFun :=
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                (_ :
                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                            map_smul' :=
                              (_ :
                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                    r •
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                          y) }.toMulActionHom
          y
[PROOFSTEP]
dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply, inclusion_mk, Eq.ndrec, id_eq,
  eq_mpr_eq_cast, ZeroMemClass.coe_zero, AddSubmonoid.mk_add_mk, Set.inclusion_mk]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (x y : { x // x ∈ iSup K }),
    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
        (_ :
          ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
            ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (x * y) =
      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
          (_ :
            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
              ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x *
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
          (_ :
            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
              ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) y
[PROOFSTEP]
apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))
[GOAL]
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x * y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x * Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), ↑(f i) (x * y) = ↑(f i) x * ↑(f i) y
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
on_goal 3 => rw [coe_iSup_of_directed dir]
[GOAL]
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x * y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x * Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), ↑(f i) (x * y) = ↑(f i) x * ↑(f i) y
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
on_goal 3 => rw [coe_iSup_of_directed dir]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
rw [coe_iSup_of_directed dir]
[GOAL]
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x * y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x * Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), ↑(f i) (x * y) = ↑(f i) x * ↑(f i) y
[PROOFSTEP]
all_goals simp
[GOAL]
case hopi
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x * y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x * Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
[PROOFSTEP]
simp
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x y : ↑↑(K i)), ↑(f i) (x * y) = ↑(f i) x * ↑(f i) y
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (a : { x // x ∈ iSup K }),
    MulActionHom.toFun
        {
              toDistribMulActionHom :=
                {
                  toMulActionHom :=
                    {
                      toFun :=
                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                          (_ :
                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                      map_smul' :=
                        (_ :
                          ∀ (r : R) (x : { x // x ∈ iSup K }),
                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                (_ :
                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                              r •
                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                  (_ :
                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
                  map_zero' :=
                    (_ :
                      MulActionHom.toFun
                          {
                            toFun :=
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                (_ :
                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                            map_smul' :=
                              (_ :
                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                    r •
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                          0 =
                        0),
                  map_add' :=
                    (_ :
                      ∀ (x y : { x // x ∈ iSup K }),
                        MulActionHom.toFun
                            {
                              toFun :=
                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                  (_ :
                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                              map_smul' :=
                                (_ :
                                  ∀ (r : R) (x : { x // x ∈ iSup K }),
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                      r •
                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                          (_ :
                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                            (x + y) =
                          MulActionHom.toFun
                              {
                                toFun :=
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                map_smul' :=
                                  (_ :
                                    ∀ (r : R) (x : { x // x ∈ iSup K }),
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                          (_ :
                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                        r •
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                              x +
                            MulActionHom.toFun
                              {
                                toFun :=
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                map_smul' :=
                                  (_ :
                                    ∀ (r : R) (x : { x // x ∈ iSup K }),
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                          (_ :
                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                        r •
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                              y) },
              map_mul' :=
                (_ :
                  ∀ (x y : { x // x ∈ iSup K }),
                    MulActionHom.toFun
                        {
                            toMulActionHom :=
                              {
                                toFun :=
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                map_smul' :=
                                  (_ :
                                    ∀ (r : R) (x : { x // x ∈ iSup K }),
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                          (_ :
                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                        r •
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
                            map_zero' :=
                              (_ :
                                MulActionHom.toFun
                                    {
                                      toFun :=
                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                          (_ :
                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                      map_smul' :=
                                        (_ :
                                          ∀ (r : R) (x : { x // x ∈ iSup K }),
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                              r •
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                    0 =
                                  0),
                            map_add' :=
                              (_ :
                                ∀ (x y : { x // x ∈ iSup K }),
                                  MulActionHom.toFun
                                      {
                                        toFun :=
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (x : { x // x ∈ iSup K }),
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                r •
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                      (x + y) =
                                    MulActionHom.toFun
                                        {
                                          toFun :=
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                  r •
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                        x +
                                      MulActionHom.toFun
                                        {
                                          toFun :=
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                  r •
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                        y) }.toMulActionHom
                        (x * y) =
                      MulActionHom.toFun
                          {
                              toMulActionHom :=
                                {
                                  toFun :=
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (x : { x // x ∈ iSup K }),
                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                          r •
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
                              map_zero' :=
                                (_ :
                                  MulActionHom.toFun
                                      {
                                        toFun :=
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (x : { x // x ∈ iSup K }),
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                r •
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                      0 =
                                    0),
                              map_add' :=
                                (_ :
                                  ∀ (x y : { x // x ∈ iSup K }),
                                    MulActionHom.toFun
                                        {
                                          toFun :=
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                  r •
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                        (x + y) =
                                      MulActionHom.toFun
                                          {
                                            toFun :=
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                    r •
                                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                          x +
                                        MulActionHom.toFun
                                          {
                                            toFun :=
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                    r •
                                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                          y) }.toMulActionHom
                          x *
                        MulActionHom.toFun
                          {
                              toMulActionHom :=
                                {
                                  toFun :=
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (x : { x // x ∈ iSup K }),
                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                          r •
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
                              map_zero' :=
                                (_ :
                                  MulActionHom.toFun
                                      {
                                        toFun :=
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (x : { x // x ∈ iSup K }),
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                r •
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                      0 =
                                    0),
                              map_add' :=
                                (_ :
                                  ∀ (x y : { x // x ∈ iSup K }),
                                    MulActionHom.toFun
                                        {
                                          toFun :=
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                  r •
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                        (x + y) =
                                      MulActionHom.toFun
                                          {
                                            toFun :=
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                    r •
                                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                          x +
                                        MulActionHom.toFun
                                          {
                                            toFun :=
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                    r •
                                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                          y) }.toMulActionHom
                          y) }.toDistribMulActionHom.toMulActionHom
        (star a) =
      star
        (MulActionHom.toFun
          {
                toDistribMulActionHom :=
                  {
                    toMulActionHom :=
                      {
                        toFun :=
                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                            (_ :
                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                        map_smul' :=
                          (_ :
                            ∀ (r : R) (x : { x // x ∈ iSup K }),
                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                  (_ :
                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                r •
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
                    map_zero' :=
                      (_ :
                        MulActionHom.toFun
                            {
                              toFun :=
                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                  (_ :
                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                              map_smul' :=
                                (_ :
                                  ∀ (r : R) (x : { x // x ∈ iSup K }),
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                      r •
                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                          (_ :
                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                            0 =
                          0),
                    map_add' :=
                      (_ :
                        ∀ (x y : { x // x ∈ iSup K }),
                          MulActionHom.toFun
                              {
                                toFun :=
                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                    (_ :
                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                map_smul' :=
                                  (_ :
                                    ∀ (r : R) (x : { x // x ∈ iSup K }),
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                          (_ :
                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                        r •
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                              (x + y) =
                            MulActionHom.toFun
                                {
                                  toFun :=
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (x : { x // x ∈ iSup K }),
                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                          r •
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                x +
                              MulActionHom.toFun
                                {
                                  toFun :=
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (x : { x // x ∈ iSup K }),
                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                          r •
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                y) },
                map_mul' :=
                  (_ :
                    ∀ (x y : { x // x ∈ iSup K }),
                      MulActionHom.toFun
                          {
                              toMulActionHom :=
                                {
                                  toFun :=
                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                      (_ :
                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                  map_smul' :=
                                    (_ :
                                      ∀ (r : R) (x : { x // x ∈ iSup K }),
                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                          r •
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
                              map_zero' :=
                                (_ :
                                  MulActionHom.toFun
                                      {
                                        toFun :=
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                            (_ :
                                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                (hxj : x ∈ (fun i => ↑(K i)) j),
                                                (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                  (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                        map_smul' :=
                                          (_ :
                                            ∀ (r : R) (x : { x // x ∈ iSup K }),
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                r •
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                      0 =
                                    0),
                              map_add' :=
                                (_ :
                                  ∀ (x y : { x // x ∈ iSup K }),
                                    MulActionHom.toFun
                                        {
                                          toFun :=
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                  r •
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                        (x + y) =
                                      MulActionHom.toFun
                                          {
                                            toFun :=
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                    r •
                                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                          x +
                                        MulActionHom.toFun
                                          {
                                            toFun :=
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                    r •
                                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                          y) }.toMulActionHom
                          (x * y) =
                        MulActionHom.toFun
                            {
                                toMulActionHom :=
                                  {
                                    toFun :=
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                    map_smul' :=
                                      (_ :
                                        ∀ (r : R) (x : { x // x ∈ iSup K }),
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                            r •
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
                                map_zero' :=
                                  (_ :
                                    MulActionHom.toFun
                                        {
                                          toFun :=
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                  r •
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                        0 =
                                      0),
                                map_add' :=
                                  (_ :
                                    ∀ (x y : { x // x ∈ iSup K }),
                                      MulActionHom.toFun
                                          {
                                            toFun :=
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                    r •
                                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                          (x + y) =
                                        MulActionHom.toFun
                                            {
                                              toFun :=
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                      r •
                                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                          (_ :
                                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                            x +
                                          MulActionHom.toFun
                                            {
                                              toFun :=
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                      r •
                                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                          (_ :
                                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                            y) }.toMulActionHom
                            x *
                          MulActionHom.toFun
                            {
                                toMulActionHom :=
                                  {
                                    toFun :=
                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                        (_ :
                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                    map_smul' :=
                                      (_ :
                                        ∀ (r : R) (x : { x // x ∈ iSup K }),
                                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                            r •
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) },
                                map_zero' :=
                                  (_ :
                                    MulActionHom.toFun
                                        {
                                          toFun :=
                                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                              (_ :
                                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                  (hxj : x ∈ (fun i => ↑(K i)) j),
                                                  (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                    (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                          map_smul' :=
                                            (_ :
                                              ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                    (_ :
                                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                        (hxj : x ∈ (fun i => ↑(K i)) j),
                                                        (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                          (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                  r •
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                        0 =
                                      0),
                                map_add' :=
                                  (_ :
                                    ∀ (x y : { x // x ∈ iSup K }),
                                      MulActionHom.toFun
                                          {
                                            toFun :=
                                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                (_ :
                                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                    (hxj : x ∈ (fun i => ↑(K i)) j),
                                                    (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                      (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                            map_smul' :=
                                              (_ :
                                                ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                      (_ :
                                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                          (hxj : x ∈ (fun i => ↑(K i)) j),
                                                          (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                            (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                    r •
                                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                          (x + y) =
                                        MulActionHom.toFun
                                            {
                                              toFun :=
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                      r •
                                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                          (_ :
                                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                            x +
                                          MulActionHom.toFun
                                            {
                                              toFun :=
                                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                  (_ :
                                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                      (hxj : x ∈ (fun i => ↑(K i)) j),
                                                      (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                        (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)),
                                              map_smul' :=
                                                (_ :
                                                  ∀ (r : R) (x : { x // x ∈ iSup K }),
                                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                        (_ :
                                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                            (hxj : x ∈ (fun i => ↑(K i)) j),
                                                            (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                              (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (r • x) =
                                                      r •
                                                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
                                                          (_ :
                                                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)
                                                              (hxj : x ∈ (fun i => ↑(K i)) j),
                                                              (fun i x => ↑(f i) x) i { val := x, property := hxi } =
                                                                (fun i x => ↑(f i) x) j { val := x, property := hxj })
                                                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x) }
                                            y) }.toMulActionHom
                            y) }.toDistribMulActionHom.toMulActionHom
          a)
[PROOFSTEP]
dsimp only [SetLike.coe_sort_coe, NonUnitalStarAlgHom.comp_apply, inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast,
  ZeroMemClass.coe_zero, AddSubmonoid.mk_add_mk, Set.inclusion_mk, MulMemClass.mk_mul_mk,
  NonUnitalAlgHom.toDistribMulActionHom_eq_coe, DistribMulActionHom.toFun_eq_coe,
  NonUnitalAlgHom.coe_to_distribMulActionHom, NonUnitalAlgHom.coe_mk]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (a : { x // x ∈ iSup K }),
    Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
        (_ :
          ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
            ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (star a) =
      star
        (Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
          (_ :
            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
              ↑(f i) { val := x, property := hxi } = ↑(f j) { val := x, property := hxj })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) a)
[PROOFSTEP]
apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => star x) (fun _ _ => rfl)
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x : ↑↑(K i)), ↑(f i) (star x) = star (↑(f i) x)
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
on_goal 2 => rw [coe_iSup_of_directed dir]
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x : ↑↑(K i)), ↑(f i) (star x) = star (↑(f i) x)
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
on_goal 2 => rw [coe_iSup_of_directed dir]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)
[PROOFSTEP]
rw [coe_iSup_of_directed dir]
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x : ↑↑(K i)), ↑(f i) (star x) = star (↑(f i) x)
[PROOFSTEP]
all_goals simp [map_star]
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (i : ι) (x : ↑↑(K i)), ↑(f i) (star x) = star (↑(f i) x)
[PROOFSTEP]
simp [map_star]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
x : { x // x ∈ K i }
h : K i ≤ T
⊢ ↑(iSupLift K dir f hf T hT) (↑(inclusion h) x) = ↑(f i) x
[PROOFSTEP]
subst T
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i : ι
x : { x // x ∈ K i }
h : K i ≤ iSup K
⊢ ↑(iSupLift K dir f hf (iSup K) (_ : iSup K = iSup K)) (↑(inclusion h) x) = ↑(f i) x
[PROOFSTEP]
dsimp [iSupLift]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i : ι
x : { x // x ∈ K i }
h : K i ≤ iSup K
⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
      (_ :
        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
          (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj })
      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) (↑(inclusion h) x) =
    ↑(f i) x
[PROOFSTEP]
apply Set.iUnionLift_inclusion
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i : ι
x : { x // x ∈ K i }
h : K i ≤ iSup K
⊢ ↑(K i) ⊆ ↑(iSup K)
[PROOFSTEP]
exact h
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
h : K i ≤ T
⊢ NonUnitalStarAlgHom.comp (iSupLift K dir f hf T hT) (inclusion h) = f i
[PROOFSTEP]
ext
[GOAL]
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
h : K i ≤ T
x✝ : { x // x ∈ K i }
⊢ ↑(NonUnitalStarAlgHom.comp (iSupLift K dir f hf T hT) (inclusion h)) x✝ = ↑(f i) x✝
[PROOFSTEP]
simp
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
x : { x // x ∈ K i }
hx : ↑x ∈ T
⊢ ↑(iSupLift K dir f hf T hT) { val := ↑x, property := hx } = ↑(f i) x
[PROOFSTEP]
subst hT
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i : ι
x : { x // x ∈ K i }
hx : ↑x ∈ iSup K
⊢ ↑(iSupLift K dir f hf (iSup K) (_ : iSup K = iSup K)) { val := ↑x, property := hx } = ↑(f i) x
[PROOFSTEP]
dsimp [iSupLift]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i : ι
x : { x // x ∈ K i }
hx : ↑x ∈ iSup K
⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
      (_ :
        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
          (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj })
      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) { val := ↑x, property := hx } =
    ↑(f i) x
[PROOFSTEP]
apply Set.iUnionLift_mk
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
x : { x // x ∈ T }
hx : ↑x ∈ K i
⊢ ↑(iSupLift K dir f hf T hT) x = ↑(f i) { val := ↑x, property := hx }
[PROOFSTEP]
subst hT
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i : ι
x : { x // x ∈ iSup K }
hx : ↑x ∈ K i
⊢ ↑(iSupLift K dir f hf (iSup K) (_ : iSup K = iSup K)) x = ↑(f i) { val := ↑x, property := hx }
[PROOFSTEP]
dsimp [iSupLift]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → { x // x ∈ K i } →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i : ι
x : { x // x ∈ iSup K }
hx : ↑x ∈ K i
⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)
      (_ :
        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),
          (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj })
      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ (i : ι), ↑(K i)) x =
    ↑(f i) { val := ↑x, property := hx }
[PROOFSTEP]
apply Set.iUnionLift_of_mem
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
s : Set A
z : A
⊢ z ∈ centralizer R s ↔ ∀ (g : A), g ∈ s → g * z = z * g ∧ star g * z = z * star g
[PROOFSTEP]
show (∀ g ∈ s ∪ star s, g * z = z * g) ↔ ∀ g ∈ s, g * z = z * g ∧ star g * z = z * star g
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
s : Set A
z : A
⊢ (∀ (g : A), g ∈ s ∪ star s → g * z = z * g) ↔ ∀ (g : A), g ∈ s → g * z = z * g ∧ star g * z = z * star g
[PROOFSTEP]
simp only [Set.mem_union, or_imp, forall_and, and_congr_right_iff]
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
s : Set A
z : A
⊢ (∀ (x : A), x ∈ s → x * z = z * x) →
    ((∀ (x : A), x ∈ star s → x * z = z * x) ↔ ∀ (x : A), x ∈ s → star x * z = z * star x)
[PROOFSTEP]
exact fun _ => ⟨fun hz a ha => hz _ (Set.star_mem_star.mpr ha), fun hz a ha => star_star a ▸ hz _ ha⟩
[GOAL]
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
⊢ ↑(centralizer R Set.univ) = ↑(center R A)
[PROOFSTEP]
rw [coe_centralizer, Set.univ_union, coe_center, Set.centralizer_univ]