[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ #((n : ℕ) × Functions (Language.sum L (skolem₁ L)) n) = #((n : ℕ) × BoundedFormula L Empty (n + 1))
[PROOFSTEP]
simp only [card_functions_sum, skolem₁_Functions, mk_sigma, sum_add_distrib']
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ ((sum fun i => lift #(Functions L i)) + sum fun i => lift #(BoundedFormula L Empty (i + 1))) =
    sum fun i => #(BoundedFormula L Empty (i + 1))
[PROOFSTEP]
conv_lhs => enter [2, 1, i]; rw [lift_id'.{u, v}]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
| (sum fun i => lift #(Functions L i)) + sum fun i => lift #(BoundedFormula L Empty (i + 1))
[PROOFSTEP]
enter [2, 1, i]; rw [lift_id'.{u, v}]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
| (sum fun i => lift #(Functions L i)) + sum fun i => lift #(BoundedFormula L Empty (i + 1))
[PROOFSTEP]
enter [2, 1, i]; rw [lift_id'.{u, v}]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
| (sum fun i => lift #(Functions L i)) + sum fun i => lift #(BoundedFormula L Empty (i + 1))
[PROOFSTEP]
enter [2, 1, i]
[GOAL]
case h
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
i : ℕ
| lift #(BoundedFormula L Empty (i + 1))
[PROOFSTEP]
rw [lift_id'.{u, v}]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ ((sum fun i => lift #(Functions L i)) + sum fun i => #(BoundedFormula L Empty (i + 1))) =
    sum fun i => #(BoundedFormula L Empty (i + 1))
[PROOFSTEP]
rw [add_comm, add_eq_max, max_eq_left]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ (sum fun i => lift #(Functions L i)) ≤ sum fun i => #(BoundedFormula L Empty (i + 1))
[PROOFSTEP]
refine' sum_le_sum _ _ fun n => _
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
n : ℕ
⊢ lift #(Functions L n) ≤ #(BoundedFormula L Empty (n + 1))
[PROOFSTEP]
rw [← lift_le.{_, max u v}, lift_lift, lift_mk_le.{v}]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
n : ℕ
⊢ Nonempty (Functions L n ↪ BoundedFormula L Empty (n + 1))
[PROOFSTEP]
refine' ⟨⟨fun f => (func f default).bdEqual (func f default), fun f g h => _⟩⟩
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
n : ℕ
f g : Functions L n
h : (fun f => func f default =' func f default) f = (fun f => func f default =' func f default) g
⊢ f = g
[PROOFSTEP]
rcases h with ⟨rfl, ⟨rfl⟩⟩
[GOAL]
case refl
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
n : ℕ
f : Functions L n
⊢ f = f
[PROOFSTEP]
rfl
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ ℵ₀ ≤ sum fun i => #(BoundedFormula L Empty (i + 1))
[PROOFSTEP]
rw [← mk_sigma]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ ℵ₀ ≤ #((i : ℕ) × BoundedFormula L Empty (i + 1))
[PROOFSTEP]
exact infinite_iff.1 (Infinite.of_injective (fun n => ⟨n, ⊥⟩) fun x y xy => (Sigma.mk.inj_iff.1 xy).1)
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ #((n : ℕ) × Functions (Language.sum L (skolem₁ L)) n) ≤ max ℵ₀ (card L)
[PROOFSTEP]
rw [card_functions_sum_skolem₁]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ #((n : ℕ) × BoundedFormula L Empty (n + 1)) ≤ max ℵ₀ (card L)
[PROOFSTEP]
trans #(Σ n, L.BoundedFormula Empty n)
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ #((n : ℕ) × BoundedFormula L Empty (n + 1)) ≤ #((n : ℕ) × BoundedFormula L Empty n)
[PROOFSTEP]
exact ⟨⟨Sigma.map Nat.succ fun _ => id, Nat.succ_injective.sigma_map fun _ => Function.injective_id⟩⟩
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ #((n : ℕ) × BoundedFormula L Empty n) ≤ max ℵ₀ (card L)
[PROOFSTEP]
refine' _root_.trans BoundedFormula.card_le (lift_le.{max u v}.1 _)
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ lift (max ℵ₀ (lift #Empty + lift (card L))) ≤ lift (max ℵ₀ (card L))
[PROOFSTEP]
simp only [mk_empty, lift_zero, lift_uzero, zero_add]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ lift (max ℵ₀ (card L)) ≤ lift (max ℵ₀ (card L))
[PROOFSTEP]
rfl
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
S : Substructure (Language.sum L (skolem₁ L)) M
⊢ IsElementary (↑(LHom.substructureReduct LHom.sumInl) S)
[PROOFSTEP]
apply (LHom.sumInl.substructureReduct S).isElementary_of_exists
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
S : Substructure (Language.sum L (skolem₁ L)) M
⊢ ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S })
    (a : M),
    BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a) →
      ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)
[PROOFSTEP]
intro n φ x a h
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
S : Substructure (Language.sum L (skolem₁ L)) M
n : ℕ
φ : BoundedFormula L Empty (n + 1)
x : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }
a : M
h : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)
⊢ ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)
[PROOFSTEP]
let φ' : (L.sum L.skolem₁).Functions n := LHom.sumInr.onFunction φ
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
S : Substructure (Language.sum L (skolem₁ L)) M
n : ℕ
φ : BoundedFormula L Empty (n + 1)
x : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }
a : M
h : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)
φ' : Functions (Language.sum L (skolem₁ L)) n := LHom.onFunction LHom.sumInr φ
⊢ ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)
[PROOFSTEP]
exact
  ⟨⟨funMap φ' ((↑) ∘ x), S.fun_mem (LHom.sumInr.onFunction φ) ((↑) ∘ x) (by exact fun i => (x i).2)⟩, by
    exact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)) ⟨a, h⟩⟩
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
S : Substructure (Language.sum L (skolem₁ L)) M
n : ℕ
φ : BoundedFormula L Empty (n + 1)
x : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }
a : M
h : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)
φ' : Functions (Language.sum L (skolem₁ L)) n := LHom.onFunction LHom.sumInr φ
⊢ ∀ (i : Fin n), (Subtype.val ∘ x) i ∈ ↑S
[PROOFSTEP]
exact fun i => (x i).2
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
S : Substructure (Language.sum L (skolem₁ L)) M
n : ℕ
φ : BoundedFormula L Empty (n + 1)
x : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }
a : M
h : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)
φ' : Functions (Language.sum L (skolem₁ L)) n := LHom.onFunction LHom.sumInr φ
⊢ BoundedFormula.Realize φ default
    (Fin.snoc (Subtype.val ∘ x)
      ↑{ val := funMap φ' (Subtype.val ∘ x),
          property := (_ : funMap (LHom.onFunction LHom.sumInr φ) (Subtype.val ∘ x) ∈ ↑S) })
[PROOFSTEP]
exact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)) ⟨a, h⟩
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ Small.{?u.38330, w} { x // x ∈ elementarySkolem₁Reduct ⊥ }
[PROOFSTEP]
rw [coeSort_elementarySkolem₁Reduct]
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
⊢ Small.{?u.38330, w} { x // x ∈ ⊥ }
[PROOFSTEP]
infer_instance
[GOAL]
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h1 : ℵ₀ ≤ κ
h2 : lift #↑s ≤ lift κ
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
⊢ ∃ S, s ⊆ ↑S ∧ lift #{ x // x ∈ S } = lift κ
[PROOFSTEP]
obtain ⟨s', hs'⟩ := Cardinal.le_mk_iff_exists_set.1 h4
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h1 : ℵ₀ ≤ κ
h2 : lift #↑s ≤ lift κ
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
hs' : #↑s' = lift κ
⊢ ∃ S, s ⊆ ↑S ∧ lift #{ x // x ∈ S } = lift κ
[PROOFSTEP]
rw [← aleph0_le_lift.{_, w}] at h1
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h1 : ℵ₀ ≤ lift κ
h2 : lift #↑s ≤ lift κ
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
hs' : #↑s' = lift κ
⊢ ∃ S, s ⊆ ↑S ∧ lift #{ x // x ∈ S } = lift κ
[PROOFSTEP]
rw [← hs'] at h1 h2 ⊢
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
⊢ ∃ S, s ⊆ ↑S ∧ lift #{ x // x ∈ S } = #↑s'
[PROOFSTEP]
refine'
  ⟨elementarySkolem₁Reduct (closure (L.sum L.skolem₁) (s ∪ Equiv.ulift '' s')),
    (s.subset_union_left _).trans subset_closure, _⟩
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
⊢ lift
      #{ x //
          x ∈
            elementarySkolem₁Reduct
              (LowerAdjoint.toFun (closure (Language.sum L (skolem₁ L))) (s ∪ ↑Equiv.ulift '' s')) } =
    #↑s'
[PROOFSTEP]
have h := mk_image_eq_lift _ s' Equiv.ulift.injective
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = lift #↑s'
⊢ lift
      #{ x //
          x ∈
            elementarySkolem₁Reduct
              (LowerAdjoint.toFun (closure (Language.sum L (skolem₁ L))) (s ∪ ↑Equiv.ulift '' s')) } =
    #↑s'
[PROOFSTEP]
rw [lift_umax.{w, w'}, lift_id'.{w, w'}] at h
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ lift
      #{ x //
          x ∈
            elementarySkolem₁Reduct
              (LowerAdjoint.toFun (closure (Language.sum L (skolem₁ L))) (s ∪ ↑Equiv.ulift '' s')) } =
    #↑s'
[PROOFSTEP]
rw [coeSort_elementarySkolem₁Reduct, ← h, lift_inj]
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ #{ x // x ∈ LowerAdjoint.toFun (closure (Language.sum L (skolem₁ L))) (s ∪ ↑Equiv.ulift '' s') } =
    #↑(↑Equiv.ulift '' s')
[PROOFSTEP]
refine'
  le_antisymm (lift_le.1 (lift_card_closure_le.trans _))
    (mk_le_mk_of_subset ((Set.subset_union_right _ _).trans subset_closure))
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ max ℵ₀ (lift #↑(s ∪ ↑Equiv.ulift '' s') + lift #((i : ℕ) × Functions (Language.sum L (skolem₁ L)) i)) ≤
    lift #↑(↑Equiv.ulift '' s')
[PROOFSTEP]
rw [max_le_iff, aleph0_le_lift, ← aleph0_le_lift.{_, w'}, h, add_eq_max, max_le_iff, lift_le]
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ ℵ₀ ≤ #↑s' ∧
    #↑(s ∪ ↑Equiv.ulift '' s') ≤ #↑(↑Equiv.ulift '' s') ∧
      lift #((i : ℕ) × Functions (Language.sum L (skolem₁ L)) i) ≤ lift #↑(↑Equiv.ulift '' s')
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ ℵ₀ ≤ lift #↑(s ∪ ↑Equiv.ulift '' s')
[PROOFSTEP]
refine' ⟨h1, (mk_union_le _ _).trans _, (lift_le.2 card_functions_sum_skolem₁_le).trans _⟩
[GOAL]
case intro.refine'_1
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ #↑s + #↑(↑Equiv.ulift '' s') ≤ #↑(↑Equiv.ulift '' s')
[PROOFSTEP]
rw [← lift_le, lift_add, h, add_comm, add_eq_max h1]
[GOAL]
case intro.refine'_1
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ max (#↑s') (lift #↑s) ≤ #↑s'
[PROOFSTEP]
exact max_le le_rfl h2
[GOAL]
case intro.refine'_2
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ lift (max ℵ₀ (card L)) ≤ lift #↑(↑Equiv.ulift '' s')
[PROOFSTEP]
rw [lift_max, lift_aleph0, max_le_iff, aleph0_le_lift, and_comm, ← lift_le.{w'}, lift_lift, lift_lift, ← aleph0_le_lift,
  h]
[GOAL]
case intro.refine'_2
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ lift (card L) ≤ lift #↑(↑Equiv.ulift '' s') ∧ ℵ₀ ≤ #↑s'
[PROOFSTEP]
refine' ⟨_, h1⟩
[GOAL]
case intro.refine'_2
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ lift (card L) ≤ lift #↑(↑Equiv.ulift '' s')
[PROOFSTEP]
rw [← lift_lift.{w', w}]
[GOAL]
case intro.refine'_2
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ lift (lift (card L)) ≤ lift #↑(↑Equiv.ulift '' s')
[PROOFSTEP]
refine' _root_.trans (lift_le.{w}.2 h3) _
[GOAL]
case intro.refine'_2
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ lift (lift κ) ≤ lift #↑(↑Equiv.ulift '' s')
[PROOFSTEP]
rw [lift_lift, ← lift_lift.{w, max u v}, ← hs', ← h, lift_lift]
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ ℵ₀ ≤ lift #↑(s ∪ ↑Equiv.ulift '' s')
[PROOFSTEP]
refine' _root_.trans _ (lift_le.2 (mk_le_mk_of_subset (Set.subset_union_right _ _)))
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ ℵ₀ ≤ lift #↑(↑Equiv.ulift '' s')
[PROOFSTEP]
rw [aleph0_le_lift, ← aleph0_le_lift, h]
[GOAL]
case intro
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
s : Set M
κ : Cardinal.{w'}
h3 : lift (card L) ≤ lift κ
h4 : lift κ ≤ lift #M
s' : Set (ULift M)
h2 : lift #↑s ≤ #↑s'
h1 : ℵ₀ ≤ #↑s'
hs' : #↑s' = lift κ
h : lift #↑(↑Equiv.ulift '' s') = #↑s'
⊢ ℵ₀ ≤ #↑s'
[PROOFSTEP]
exact h1


[GOAL]
D : GlueData
i j : D.J
⊢ LocallyRingedSpace.IsOpenImmersion (f (toLocallyRingedSpaceGlueData D).toGlueData i j)
[PROOFSTEP]
apply GlueData.f_open
[GOAL]
D : GlueData
i j : D.J
⊢ SheafedSpace.IsOpenImmersion
    (f (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)).toGlueData i j)
[PROOFSTEP]
apply GlueData.f_open
[GOAL]
D : GlueData
i j : D.J
⊢ PresheafedSpace.IsOpenImmersion
    (f
      (SheafedSpace.GlueData.toPresheafedSpaceGlueData
          (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D))).toGlueData
      i j)
[PROOFSTEP]
apply GlueData.f_open
[GOAL]
D : GlueData
i : D.J
⊢ LocallyRingedSpace.IsOpenImmersion (ι (toLocallyRingedSpaceGlueData D).toGlueData i)
[PROOFSTEP]
apply LocallyRingedSpace.GlueData.ι_isOpenImmersion
[GOAL]
D : GlueData
⊢ Scheme
[PROOFSTEP]
apply LocallyRingedSpace.IsOpenImmersion.scheme D.toLocallyRingedSpaceGlueData.toGlueData.glued
[GOAL]
D : GlueData
⊢ ∀ (x : ↑(LocallyRingedSpace.toTopCat (glued (toLocallyRingedSpaceGlueData D).toGlueData))),
    ∃ R f, x ∈ Set.range ↑f.val.base ∧ LocallyRingedSpace.IsOpenImmersion f
[PROOFSTEP]
intro x
[GOAL]
D : GlueData
x : ↑(LocallyRingedSpace.toTopCat (glued (toLocallyRingedSpaceGlueData D).toGlueData))
⊢ ∃ R f, x ∈ Set.range ↑f.val.base ∧ LocallyRingedSpace.IsOpenImmersion f
[PROOFSTEP]
obtain ⟨i, y, rfl⟩ := D.toLocallyRingedSpaceGlueData.ι_jointly_surjective x
[GOAL]
case intro.intro
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ ∃ R f,
    ↑(ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y ∈ Set.range ↑f.val.base ∧
      LocallyRingedSpace.IsOpenImmersion f
[PROOFSTEP]
refine' ⟨_, _ ≫ D.toLocallyRingedSpaceGlueData.toGlueData.ι i, _⟩
[GOAL]
case intro.intro.refine'_1
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ CommRingCat
case intro.intro.refine'_2
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ Spec.toLocallyRingedSpace.obj (op ?intro.intro.refine'_1) ⟶ U (toLocallyRingedSpaceGlueData D).toGlueData i
case intro.intro.refine'_3
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ ↑(ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y ∈
      Set.range ↑(?intro.intro.refine'_2 ≫ ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base ∧
    LocallyRingedSpace.IsOpenImmersion (?intro.intro.refine'_2 ≫ ι (toLocallyRingedSpaceGlueData D).toGlueData i)
[PROOFSTEP]
swap
[GOAL]
case intro.intro.refine'_2
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ Spec.toLocallyRingedSpace.obj (op ?intro.intro.refine'_1) ⟶ U (toLocallyRingedSpaceGlueData D).toGlueData i
case intro.intro.refine'_1
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ CommRingCat
case intro.intro.refine'_3
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ ↑(ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y ∈
      Set.range ↑(?intro.intro.refine'_2 ≫ ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base ∧
    LocallyRingedSpace.IsOpenImmersion (?intro.intro.refine'_2 ≫ ι (toLocallyRingedSpaceGlueData D).toGlueData i)
[PROOFSTEP]
exact (D.U i).affineCover.map y
[GOAL]
case intro.intro.refine'_3
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ ↑(ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y ∈
      Set.range
        ↑(OpenCover.map (affineCover (U D.toGlueData i)) y ≫ ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base ∧
    LocallyRingedSpace.IsOpenImmersion
      (OpenCover.map (affineCover (U D.toGlueData i)) y ≫ ι (toLocallyRingedSpaceGlueData D).toGlueData i)
[PROOFSTEP]
constructor
[GOAL]
case intro.intro.refine'_3.left
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ ↑(ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y ∈
    Set.range
      ↑(OpenCover.map (affineCover (U D.toGlueData i)) y ≫ ι (toLocallyRingedSpaceGlueData D).toGlueData i).val.base
[PROOFSTEP]
dsimp [-Set.mem_range]
[GOAL]
case intro.intro.refine'_3.left
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ ↑(ι (mapGlueData D.toGlueData forgetToLocallyRingedSpace) i).val.base y ∈
    Set.range
      ↑((OpenCover.map (affineCover (U D.toGlueData i)) y).val.base ≫
          (ι (mapGlueData D.toGlueData forgetToLocallyRingedSpace) i).val.base)
[PROOFSTEP]
rw [coe_comp, Set.range_comp]
[GOAL]
case intro.intro.refine'_3.left
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ ↑(ι (mapGlueData D.toGlueData forgetToLocallyRingedSpace) i).val.base y ∈
    ↑(ι (mapGlueData D.toGlueData forgetToLocallyRingedSpace) i).val.base ''
      Set.range ↑(OpenCover.map (affineCover (U D.toGlueData i)) y).val.base
[PROOFSTEP]
refine' Set.mem_image_of_mem _ _
[GOAL]
case intro.intro.refine'_3.left
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ y ∈ Set.range ↑(OpenCover.map (affineCover (U D.toGlueData i)) y).val.base
[PROOFSTEP]
exact (D.U i).affineCover.Covers y
[GOAL]
case intro.intro.refine'_3.right
D : GlueData
i : (toLocallyRingedSpaceGlueData D).toGlueData.J
y : ↑(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))
⊢ LocallyRingedSpace.IsOpenImmersion
    (OpenCover.map (affineCover (U D.toGlueData i)) y ≫ ι (toLocallyRingedSpaceGlueData D).toGlueData i)
[PROOFSTEP]
infer_instance
[GOAL]
D : GlueData
i : D.J
⊢ IsOpenImmersion (CategoryTheory.GlueData.ι D.toGlueData i)
[PROOFSTEP]
rw [← D.ι_isoLocallyRingedSpace_inv]
[GOAL]
D : GlueData
i : D.J
⊢ IsOpenImmersion
    (CategoryTheory.GlueData.ι (toLocallyRingedSpaceGlueData D).toGlueData i ≫ (isoLocallyRingedSpace D).inv)
[PROOFSTEP]
infer_instance
[GOAL]
D : GlueData
i j : D.J
⊢ f D.toGlueData i j ≫ ι D i = (t D.toGlueData i j ≫ f D.toGlueData j i) ≫ ι D j
[PROOFSTEP]
simp
[GOAL]
D : GlueData
⊢ ↑(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace ≅
    CategoryTheory.GlueData.glued
      (GlueData.toTopGlueData
          (SheafedSpace.GlueData.toPresheafedSpaceGlueData
            (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
[PROOFSTEP]
refine (PresheafedSpace.forget _).mapIso ?_ ≪≫ GlueData.gluedIso _ (PresheafedSpace.forget.{_, _, u} _)
[GOAL]
D : GlueData
⊢ (glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace ≅
    CategoryTheory.GlueData.glued
      (SheafedSpace.GlueData.toPresheafedSpaceGlueData
          (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D))).toGlueData
[PROOFSTEP]
refine SheafedSpace.forgetToPresheafedSpace.mapIso ?_ ≪≫ SheafedSpace.GlueData.isoPresheafedSpace _
[GOAL]
D : GlueData
⊢ (glued D).toLocallyRingedSpace.toSheafedSpace ≅
    CategoryTheory.GlueData.glued
      (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)).toGlueData
[PROOFSTEP]
refine LocallyRingedSpace.forgetToSheafedSpace.mapIso ?_ ≪≫ LocallyRingedSpace.GlueData.isoSheafedSpace _
[GOAL]
D : GlueData
⊢ (glued D).toLocallyRingedSpace ≅ CategoryTheory.GlueData.glued (toLocallyRingedSpaceGlueData D).toGlueData
[PROOFSTEP]
exact Scheme.GlueData.isoLocallyRingedSpace _
[GOAL]
D : GlueData
i : D.J
⊢ CategoryTheory.GlueData.ι
        (GlueData.toTopGlueData
            (SheafedSpace.GlueData.toPresheafedSpaceGlueData
              (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
        i ≫
      (isoCarrier D).inv =
    (ι D i).val.base
[PROOFSTEP]
delta isoCarrier
[GOAL]
D : GlueData
i : D.J
⊢ CategoryTheory.GlueData.ι
        (GlueData.toTopGlueData
            (SheafedSpace.GlueData.toPresheafedSpaceGlueData
              (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
        i ≫
      ((PresheafedSpace.forget CommRingCat).mapIso
            (SheafedSpace.forgetToPresheafedSpace.mapIso
                (LocallyRingedSpace.forgetToSheafedSpace.mapIso (isoLocallyRingedSpace D) ≪≫
                  LocallyRingedSpace.GlueData.isoSheafedSpace (toLocallyRingedSpaceGlueData D)) ≪≫
              SheafedSpace.GlueData.isoPresheafedSpace
                (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D))) ≪≫
          gluedIso
            (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D))).toGlueData
            (PresheafedSpace.forget CommRingCat)).inv =
    (ι D i).val.base
[PROOFSTEP]
rw [Iso.trans_inv, GlueData.ι_gluedIso_inv_assoc, Functor.mapIso_inv, Iso.trans_inv, Functor.mapIso_inv, Iso.trans_inv,
  SheafedSpace.forgetToPresheafedSpace_map, forget_map, forget_map, ← comp_base, ← Category.assoc,
  D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.ι_isoPresheafedSpace_inv i]
[GOAL]
D : GlueData
i : D.J
⊢ (CategoryTheory.GlueData.ι
          (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)).toGlueData i ≫
        (LocallyRingedSpace.GlueData.isoSheafedSpace (toLocallyRingedSpaceGlueData D)).inv ≫
          (LocallyRingedSpace.forgetToSheafedSpace.mapIso (isoLocallyRingedSpace D)).inv).base =
    (ι D i).val.base
[PROOFSTEP]
erw [← Category.assoc, D.toLocallyRingedSpaceGlueData.ι_isoSheafedSpace_inv i]
[GOAL]
D : GlueData
i : D.J
⊢ ((CategoryTheory.GlueData.ι (toLocallyRingedSpaceGlueData D).toGlueData i).val ≫
        (LocallyRingedSpace.forgetToSheafedSpace.mapIso (isoLocallyRingedSpace D)).inv).base =
    (ι D i).val.base
[PROOFSTEP]
change (_ ≫ D.isoLocallyRingedSpace.inv).1.base = _
[GOAL]
D : GlueData
i : D.J
⊢ (CategoryTheory.GlueData.ι (toLocallyRingedSpaceGlueData D).toGlueData i ≫ (isoLocallyRingedSpace D).inv).val.base =
    (ι D i).val.base
[PROOFSTEP]
rw [D.ι_isoLocallyRingedSpace_inv i]
[GOAL]
D : GlueData
i j : D.J
x : ↑↑(U D.toGlueData i).toPresheafedSpace
y : ↑↑(U D.toGlueData j).toPresheafedSpace
⊢ ↑(CategoryTheory.GlueData.ι D.toGlueData i).val.base x = ↑(CategoryTheory.GlueData.ι D.toGlueData j).val.base y ↔
    Rel D { fst := i, snd := x } { fst := j, snd := y }
[PROOFSTEP]
refine'
  Iff.trans _
    (TopCat.GlueData.ι_eq_iff_rel
      D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData i j x y)
[GOAL]
D : GlueData
i j : D.J
x : ↑↑(U D.toGlueData i).toPresheafedSpace
y : ↑↑(U D.toGlueData j).toPresheafedSpace
⊢ ↑(CategoryTheory.GlueData.ι D.toGlueData i).val.base x = ↑(CategoryTheory.GlueData.ι D.toGlueData j).val.base y ↔
    ↑(CategoryTheory.GlueData.ι
            (GlueData.toTopGlueData
                (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                  (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
            i)
        x =
      ↑(CategoryTheory.GlueData.ι
            (GlueData.toTopGlueData
                (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                  (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
            j)
        y
[PROOFSTEP]
rw [← ((TopCat.mono_iff_injective D.isoCarrier.inv).mp _).eq_iff]
[GOAL]
D : GlueData
i j : D.J
x : ↑↑(U D.toGlueData i).toPresheafedSpace
y : ↑↑(U D.toGlueData j).toPresheafedSpace
⊢ ↑(CategoryTheory.GlueData.ι D.toGlueData i).val.base x = ↑(CategoryTheory.GlueData.ι D.toGlueData j).val.base y ↔
    ↑(isoCarrier D).inv
        (↑(CategoryTheory.GlueData.ι
              (GlueData.toTopGlueData
                  (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                    (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
              i)
          x) =
      ↑(isoCarrier D).inv
        (↑(CategoryTheory.GlueData.ι
              (GlueData.toTopGlueData
                  (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                    (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
              j)
          y)
[PROOFSTEP]
simp_rw [← comp_apply, ← D.ι_isoCarrier_inv]
[GOAL]
D : GlueData
i j : D.J
x : ↑↑(U D.toGlueData i).toPresheafedSpace
y : ↑↑(U D.toGlueData j).toPresheafedSpace
⊢ ↑(CategoryTheory.GlueData.ι
              (mapGlueData
                (mapGlueData
                  (mapGlueData (mapGlueData D.toGlueData forgetToLocallyRingedSpace)
                    LocallyRingedSpace.forgetToSheafedSpace)
                  SheafedSpace.forgetToPresheafedSpace)
                (PresheafedSpace.forget CommRingCat))
              i ≫
            (isoCarrier D).inv)
        x =
      ↑(CategoryTheory.GlueData.ι
              (mapGlueData
                (mapGlueData
                  (mapGlueData (mapGlueData D.toGlueData forgetToLocallyRingedSpace)
                    LocallyRingedSpace.forgetToSheafedSpace)
                  SheafedSpace.forgetToPresheafedSpace)
                (PresheafedSpace.forget CommRingCat))
              j ≫
            (isoCarrier D).inv)
        y ↔
    ↑(CategoryTheory.GlueData.ι
              (mapGlueData
                (mapGlueData
                  (mapGlueData (mapGlueData D.toGlueData forgetToLocallyRingedSpace)
                    LocallyRingedSpace.forgetToSheafedSpace)
                  SheafedSpace.forgetToPresheafedSpace)
                (PresheafedSpace.forget CommRingCat))
              i ≫
            (isoCarrier D).inv)
        x =
      ↑(CategoryTheory.GlueData.ι
              (mapGlueData
                (mapGlueData
                  (mapGlueData (mapGlueData D.toGlueData forgetToLocallyRingedSpace)
                    LocallyRingedSpace.forgetToSheafedSpace)
                  SheafedSpace.forgetToPresheafedSpace)
                (PresheafedSpace.forget CommRingCat))
              j ≫
            (isoCarrier D).inv)
        y
[PROOFSTEP]
rfl
[GOAL]
D : GlueData
i j : D.J
x : ↑↑(U D.toGlueData i).toPresheafedSpace
y : ↑↑(U D.toGlueData j).toPresheafedSpace
⊢ Mono (isoCarrier D).inv
[PROOFSTEP]
infer_instance
[GOAL]
D : GlueData
U : Set ↑↑(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
⊢ IsOpen U ↔ ∀ (i : D.J), IsOpen (↑(ι D i).val.base ⁻¹' U)
[PROOFSTEP]
rw [← (TopCat.homeoOfIso D.isoCarrier.symm).isOpen_preimage]
[GOAL]
D : GlueData
U : Set ↑↑(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
⊢ IsOpen (↑(TopCat.homeoOfIso (isoCarrier D).symm) ⁻¹' U) ↔ ∀ (i : D.J), IsOpen (↑(ι D i).val.base ⁻¹' U)
[PROOFSTEP]
rw [TopCat.GlueData.isOpen_iff]
[GOAL]
D : GlueData
U : Set ↑↑(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
⊢ (∀
      (i :
        (GlueData.toTopGlueData
              (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData.J),
      IsOpen
        (↑(CategoryTheory.GlueData.ι
              (GlueData.toTopGlueData
                  (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                    (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
              i) ⁻¹'
          (↑(TopCat.homeoOfIso (isoCarrier D).symm) ⁻¹' U))) ↔
    ∀ (i : D.J), IsOpen (↑(ι D i).val.base ⁻¹' U)
[PROOFSTEP]
apply forall_congr'
[GOAL]
case h
D : GlueData
U : Set ↑↑(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
⊢ ∀
    (a :
      (GlueData.toTopGlueData
            (SheafedSpace.GlueData.toPresheafedSpaceGlueData
              (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData.J),
    IsOpen
        (↑(CategoryTheory.GlueData.ι
              (GlueData.toTopGlueData
                  (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                    (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
              a) ⁻¹'
          (↑(TopCat.homeoOfIso (isoCarrier D).symm) ⁻¹' U)) ↔
      IsOpen (↑(ι D a).val.base ⁻¹' U)
[PROOFSTEP]
intro i
[GOAL]
case h
D : GlueData
U : Set ↑↑(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
i :
  (GlueData.toTopGlueData
        (SheafedSpace.GlueData.toPresheafedSpaceGlueData
          (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData.J
⊢ IsOpen
      (↑(CategoryTheory.GlueData.ι
            (GlueData.toTopGlueData
                (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                  (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
            i) ⁻¹'
        (↑(TopCat.homeoOfIso (isoCarrier D).symm) ⁻¹' U)) ↔
    IsOpen (↑(ι D i).val.base ⁻¹' U)
[PROOFSTEP]
erw [← Set.preimage_comp, ← ι_isoCarrier_inv]
[GOAL]
case h
D : GlueData
U : Set ↑↑(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
i :
  (GlueData.toTopGlueData
        (SheafedSpace.GlueData.toPresheafedSpaceGlueData
          (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData.J
⊢ IsOpen
      (↑(TopCat.homeoOfIso (isoCarrier D).symm) ∘
          ↑(CategoryTheory.GlueData.ι
              (GlueData.toTopGlueData
                  (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                    (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
              i) ⁻¹'
        U) ↔
    IsOpen
      (↑(CategoryTheory.GlueData.ι
              (GlueData.toTopGlueData
                  (SheafedSpace.GlueData.toPresheafedSpaceGlueData
                    (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData
              i ≫
            (isoCarrier D).inv) ⁻¹'
        U)
[PROOFSTEP]
rfl
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ pullback pullback.fst pullback.fst ⟶ pullback pullback.fst pullback.fst
[PROOFSTEP]
refine' (pullbackRightPullbackFstIso _ _ _).hom ≫ _
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ pullback (pullback.fst ≫ map 𝒰 x) (map 𝒰 z) ⟶ pullback pullback.fst pullback.fst
[PROOFSTEP]
refine' _ ≫ (pullbackSymmetry _ _).hom
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ pullback (pullback.fst ≫ map 𝒰 x) (map 𝒰 z) ⟶ pullback pullback.fst pullback.fst
[PROOFSTEP]
refine' _ ≫ (pullbackRightPullbackFstIso _ _ _).inv
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ pullback (pullback.fst ≫ map 𝒰 x) (map 𝒰 z) ⟶ pullback (pullback.fst ≫ map 𝒰 y) (map 𝒰 z)
[PROOFSTEP]
refine' pullback.map _ _ _ _ (pullbackSymmetry _ _).hom (𝟙 _) (𝟙 _) _ _
[GOAL]
case refine'_1
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ (pullback.fst ≫ map 𝒰 x) ≫ 𝟙 X = (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom ≫ pullback.fst ≫ map 𝒰 y
[PROOFSTEP]
simp [pullback.condition]
[GOAL]
case refine'_2
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ map 𝒰 z ≫ 𝟙 X = 𝟙 (obj 𝒰 z) ≫ map 𝒰 z
[PROOFSTEP]
simp
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd
[PROOFSTEP]
delta gluedCoverT'
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ ((pullbackRightPullbackFstIso (map 𝒰 x) (map 𝒰 z) pullback.fst).hom ≫
        (pullback.map (pullback.fst ≫ map 𝒰 x) (map 𝒰 z) (pullback.fst ≫ map 𝒰 y) (map 𝒰 z)
              (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom (𝟙 (obj 𝒰 z)) (𝟙 X)
              (_ : (pullback.fst ≫ map 𝒰 x) ≫ 𝟙 X = (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom ≫ pullback.fst ≫ map 𝒰 y)
              (_ : map 𝒰 z ≫ 𝟙 X = 𝟙 (obj 𝒰 z) ≫ map 𝒰 z) ≫
            (pullbackRightPullbackFstIso (map 𝒰 y) (map 𝒰 z) pullback.fst).inv) ≫
          (pullbackSymmetry pullback.fst pullback.fst).hom) ≫
      pullback.fst ≫ pullback.fst =
    pullback.fst ≫ pullback.snd
[PROOFSTEP]
simp
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ pullback.fst ≫ pullback.snd = pullback.snd ≫ pullback.snd
[PROOFSTEP]
delta gluedCoverT'
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ ((pullbackRightPullbackFstIso (map 𝒰 x) (map 𝒰 z) pullback.fst).hom ≫
        (pullback.map (pullback.fst ≫ map 𝒰 x) (map 𝒰 z) (pullback.fst ≫ map 𝒰 y) (map 𝒰 z)
              (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom (𝟙 (obj 𝒰 z)) (𝟙 X)
              (_ : (pullback.fst ≫ map 𝒰 x) ≫ 𝟙 X = (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom ≫ pullback.fst ≫ map 𝒰 y)
              (_ : map 𝒰 z ≫ 𝟙 X = 𝟙 (obj 𝒰 z) ≫ map 𝒰 z) ≫
            (pullbackRightPullbackFstIso (map 𝒰 y) (map 𝒰 z) pullback.fst).inv) ≫
          (pullbackSymmetry pullback.fst pullback.fst).hom) ≫
      pullback.fst ≫ pullback.snd =
    pullback.snd ≫ pullback.snd
[PROOFSTEP]
simp
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ pullback.snd
[PROOFSTEP]
delta gluedCoverT'
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ ((pullbackRightPullbackFstIso (map 𝒰 x) (map 𝒰 z) pullback.fst).hom ≫
        (pullback.map (pullback.fst ≫ map 𝒰 x) (map 𝒰 z) (pullback.fst ≫ map 𝒰 y) (map 𝒰 z)
              (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom (𝟙 (obj 𝒰 z)) (𝟙 X)
              (_ : (pullback.fst ≫ map 𝒰 x) ≫ 𝟙 X = (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom ≫ pullback.fst ≫ map 𝒰 y)
              (_ : map 𝒰 z ≫ 𝟙 X = 𝟙 (obj 𝒰 z) ≫ map 𝒰 z) ≫
            (pullbackRightPullbackFstIso (map 𝒰 y) (map 𝒰 z) pullback.fst).inv) ≫
          (pullbackSymmetry pullback.fst pullback.fst).hom) ≫
      pullback.snd ≫ pullback.fst =
    pullback.fst ≫ pullback.snd
[PROOFSTEP]
simp
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ pullback.snd ≫ pullback.snd = pullback.fst ≫ pullback.fst
[PROOFSTEP]
delta gluedCoverT'
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ ((pullbackRightPullbackFstIso (map 𝒰 x) (map 𝒰 z) pullback.fst).hom ≫
        (pullback.map (pullback.fst ≫ map 𝒰 x) (map 𝒰 z) (pullback.fst ≫ map 𝒰 y) (map 𝒰 z)
              (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom (𝟙 (obj 𝒰 z)) (𝟙 X)
              (_ : (pullback.fst ≫ map 𝒰 x) ≫ 𝟙 X = (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom ≫ pullback.fst ≫ map 𝒰 y)
              (_ : map 𝒰 z ≫ 𝟙 X = 𝟙 (obj 𝒰 z) ≫ map 𝒰 z) ≫
            (pullbackRightPullbackFstIso (map 𝒰 y) (map 𝒰 z) pullback.fst).inv) ≫
          (pullbackSymmetry pullback.fst pullback.fst).hom) ≫
      pullback.snd ≫ pullback.snd =
    pullback.fst ≫ pullback.fst
[PROOFSTEP]
simp
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.fst = pullback.fst
[PROOFSTEP]
apply pullback.hom_ext
[GOAL]
case h₀
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.fst) ≫ pullback.fst =
    pullback.fst ≫ pullback.fst
[PROOFSTEP]
simp
[GOAL]
case h₁
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.fst) ≫ pullback.snd =
    pullback.fst ≫ pullback.snd
[PROOFSTEP]
simp
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.snd = pullback.snd
[PROOFSTEP]
apply pullback.hom_ext
[GOAL]
case h₀
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.snd) ≫ pullback.fst =
    pullback.snd ≫ pullback.fst
[PROOFSTEP]
simp [pullback.condition]
[GOAL]
case h₁
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.snd) ≫ pullback.snd =
    pullback.snd ≫ pullback.snd
[PROOFSTEP]
simp [pullback.condition]
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y = 𝟙 (pullback pullback.fst pullback.fst)
[PROOFSTEP]
apply pullback.hom_ext
[GOAL]
case h₀
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y) ≫ pullback.fst =
    𝟙 (pullback pullback.fst pullback.fst) ≫ pullback.fst
[PROOFSTEP]
simp_rw [Category.id_comp, Category.assoc]
[GOAL]
case h₁
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y) ≫ pullback.snd =
    𝟙 (pullback pullback.fst pullback.fst) ≫ pullback.snd
[PROOFSTEP]
simp_rw [Category.id_comp, Category.assoc]
[GOAL]
case h₀
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.fst = pullback.fst
case h₁
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.snd = pullback.snd
[PROOFSTEP]
apply glued_cover_cocycle_fst
[GOAL]
case h₁
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.snd = pullback.snd
[PROOFSTEP]
apply glued_cover_cocycle_snd
[GOAL]
X : Scheme
𝒰 : OpenCover X
x : 𝒰.J
⊢ (fun x y => (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom) x x =
    𝟙
      ((fun x =>
          match x with
          | (x, y) => pullback (map 𝒰 x) (map 𝒰 y))
        (x, x))
[PROOFSTEP]
simp
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ (fun x y z => gluedCoverT' 𝒰 x y z) x y z ≫ pullback.snd =
    pullback.fst ≫ (fun x y => (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom) x y
[PROOFSTEP]
apply pullback.hom_ext
[GOAL]
case h₀
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ ((fun x y z => gluedCoverT' 𝒰 x y z) x y z ≫ pullback.snd) ≫ pullback.fst =
    (pullback.fst ≫ (fun x y => (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom) x y) ≫ pullback.fst
[PROOFSTEP]
simp
  -- The `cocycle` field could have been `by tidy` but lean timeouts.
[GOAL]
case h₁
X : Scheme
𝒰 : OpenCover X
x y z : 𝒰.J
⊢ ((fun x y z => gluedCoverT' 𝒰 x y z) x y z ≫ pullback.snd) ≫ pullback.snd =
    (pullback.fst ≫ (fun x y => (pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom) x y) ≫ pullback.snd
[PROOFSTEP]
simp
  -- The `cocycle` field could have been `by tidy` but lean timeouts.
[GOAL]
X : Scheme
𝒰 : OpenCover X
⊢ GlueData.glued (gluedCover 𝒰) ⟶ X
[PROOFSTEP]
fapply Multicoequalizer.desc
[GOAL]
case k
X : Scheme
𝒰 : OpenCover X
⊢ (b : (diagram (gluedCover 𝒰).toGlueData).R) → MultispanIndex.right (diagram (gluedCover 𝒰).toGlueData) b ⟶ X
case h
X : Scheme
𝒰 : OpenCover X
⊢ ∀ (a : (diagram (gluedCover 𝒰).toGlueData).L),
    MultispanIndex.fst (diagram (gluedCover 𝒰).toGlueData) a ≫
        ?k (MultispanIndex.fstFrom (diagram (gluedCover 𝒰).toGlueData) a) =
      MultispanIndex.snd (diagram (gluedCover 𝒰).toGlueData) a ≫
        ?k (MultispanIndex.sndFrom (diagram (gluedCover 𝒰).toGlueData) a)
[PROOFSTEP]
exact fun x => 𝒰.map x
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
⊢ ∀ (a : (diagram (gluedCover 𝒰).toGlueData).L),
    MultispanIndex.fst (diagram (gluedCover 𝒰).toGlueData) a ≫
        map 𝒰 (MultispanIndex.fstFrom (diagram (gluedCover 𝒰).toGlueData) a) =
      MultispanIndex.snd (diagram (gluedCover 𝒰).toGlueData) a ≫
        map 𝒰 (MultispanIndex.sndFrom (diagram (gluedCover 𝒰).toGlueData) a)
[PROOFSTEP]
rintro ⟨x, y⟩
[GOAL]
case h.mk
X : Scheme
𝒰 : OpenCover X
x y : (gluedCover 𝒰).toGlueData.J
⊢ MultispanIndex.fst (diagram (gluedCover 𝒰).toGlueData) (x, y) ≫
      map 𝒰 (MultispanIndex.fstFrom (diagram (gluedCover 𝒰).toGlueData) (x, y)) =
    MultispanIndex.snd (diagram (gluedCover 𝒰).toGlueData) (x, y) ≫
      map 𝒰 (MultispanIndex.sndFrom (diagram (gluedCover 𝒰).toGlueData) (x, y))
[PROOFSTEP]
change pullback.fst ≫ _ = ((pullbackSymmetry _ _).hom ≫ pullback.fst) ≫ _
[GOAL]
case h.mk
X : Scheme
𝒰 : OpenCover X
x y : (gluedCover 𝒰).toGlueData.J
⊢ pullback.fst ≫ map 𝒰 (MultispanIndex.fstFrom (diagram (gluedCover 𝒰).toGlueData) (x, y)) =
    ((pullbackSymmetry (map 𝒰 x) (map 𝒰 y)).hom ≫ pullback.fst) ≫
      map 𝒰 (MultispanIndex.sndFrom (diagram (gluedCover 𝒰).toGlueData) (x, y))
[PROOFSTEP]
simpa using pullback.condition
[GOAL]
X : Scheme
𝒰 : OpenCover X
⊢ Function.Injective ↑(fromGlued 𝒰).val.base
[PROOFSTEP]
intro x y h
[GOAL]
X : Scheme
𝒰 : OpenCover X
x y :
  (CategoryTheory.forget TopCat).obj
    ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
h : ↑(fromGlued 𝒰).val.base x = ↑(fromGlued 𝒰).val.base y
⊢ x = y
[PROOFSTEP]
obtain ⟨i, x, rfl⟩ := 𝒰.gluedCover.ι_jointly_surjective x
[GOAL]
case intro.intro
X : Scheme
𝒰 : OpenCover X
y :
  (CategoryTheory.forget TopCat).obj
    ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
h : ↑(fromGlued 𝒰).val.base (↑(GlueData.ι (gluedCover 𝒰) i).val.base x) = ↑(fromGlued 𝒰).val.base y
⊢ ↑(GlueData.ι (gluedCover 𝒰) i).val.base x = y
[PROOFSTEP]
obtain ⟨j, y, rfl⟩ := 𝒰.gluedCover.ι_jointly_surjective y
[GOAL]
case intro.intro.intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h :
  ↑(fromGlued 𝒰).val.base (↑(GlueData.ι (gluedCover 𝒰) i).val.base x) =
    ↑(fromGlued 𝒰).val.base (↑(GlueData.ι (gluedCover 𝒰) j).val.base y)
⊢ ↑(GlueData.ι (gluedCover 𝒰) i).val.base x = ↑(GlueData.ι (gluedCover 𝒰) j).val.base y
[PROOFSTEP]
simp_rw [← comp_apply, ← SheafedSpace.comp_base, ← LocallyRingedSpace.comp_val] at h
[GOAL]
case intro.intro.intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(GlueData.ι (gluedCover 𝒰) i ≫ fromGlued 𝒰).val.base x = ↑(GlueData.ι (gluedCover 𝒰) j ≫ fromGlued 𝒰).val.base y
⊢ ↑(GlueData.ι (gluedCover 𝒰) i).val.base x = ↑(GlueData.ι (gluedCover 𝒰) j).val.base y
[PROOFSTEP]
erw [ι_fromGlued, ι_fromGlued] at h
[GOAL]
case intro.intro.intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
⊢ ↑(GlueData.ι (gluedCover 𝒰) i).val.base x = ↑(GlueData.ι (gluedCover 𝒰) j).val.base y
[PROOFSTEP]
let e :=
  (TopCat.pullbackConeIsLimit _ _).conePointUniqueUpToIso
    (isLimitOfHasPullbackOfPreservesLimit Scheme.forgetToTop (𝒰.map i) (𝒰.map j))
[GOAL]
case intro.intro.intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ↑(GlueData.ι (gluedCover 𝒰) i).val.base x = ↑(GlueData.ι (gluedCover 𝒰) j).val.base y
[PROOFSTEP]
rw [𝒰.gluedCover.ι_eq_iff]
[GOAL]
case intro.intro.intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ GlueData.Rel (gluedCover 𝒰) { fst := i, snd := x } { fst := j, snd := y }
[PROOFSTEP]
right
[GOAL]
case intro.intro.intro.intro.h
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ∃ x_1,
    ↑(GlueData.f (gluedCover 𝒰).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst).val.base x_1 =
        { fst := i, snd := x }.snd ∧
      ↑(t (gluedCover 𝒰).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst ≫
                  GlueData.f (gluedCover 𝒰).toGlueData { fst := j, snd := y }.fst { fst := i, snd := x }.fst).val.base
          x_1 =
        { fst := j, snd := y }.snd
[PROOFSTEP]
use e.hom ⟨⟨x, y⟩, h⟩
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ↑(GlueData.f (gluedCover 𝒰).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst).val.base
        (↑e.hom { val := (x, y), property := h }) =
      { fst := i, snd := x }.snd ∧
    ↑(t (gluedCover 𝒰).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst ≫
                GlueData.f (gluedCover 𝒰).toGlueData { fst := j, snd := y }.fst { fst := i, snd := x }.fst).val.base
        (↑e.hom { val := (x, y), property := h }) =
      { fst := j, snd := y }.snd
[PROOFSTEP]
constructor
  -- Porting note: in the two subproofs below, added the `change` lines
[GOAL]
case h.left
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ↑(GlueData.f (gluedCover 𝒰).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst).val.base
      (↑e.hom { val := (x, y), property := h }) =
    { fst := i, snd := x }.snd
[PROOFSTEP]
change (e.hom ≫ _) ⟨(x, y), h⟩ = x
[GOAL]
case h.left
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ↑(e.hom ≫ (GlueData.f (gluedCover 𝒰).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst).val.base)
      { val := (x, y), property := h } =
    x
[PROOFSTEP]
erw [IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingCospan.left]
[GOAL]
case h.left
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ↑(NatTrans.app (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).π WalkingCospan.left)
      { val := (x, y), property := h } =
    x
[PROOFSTEP]
rfl
[GOAL]
case h.right
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ↑(t (gluedCover 𝒰).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst ≫
              GlueData.f (gluedCover 𝒰).toGlueData { fst := j, snd := y }.fst { fst := i, snd := x }.fst).val.base
      (↑e.hom { val := (x, y), property := h }) =
    { fst := j, snd := y }.snd
[PROOFSTEP]
change (e.hom ≫ ((gluedCover 𝒰).toGlueData.t i j ≫ (gluedCover 𝒰).toGlueData.f j i).val.base) ⟨(x, y), h⟩ = y
[GOAL]
case h.right
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ↑(e.hom ≫ (t (gluedCover 𝒰).toGlueData i j ≫ GlueData.f (gluedCover 𝒰).toGlueData j i).val.base)
      { val := (x, y), property := h } =
    y
[PROOFSTEP]
erw [pullbackSymmetry_hom_comp_fst, IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingCospan.right]
[GOAL]
case h.right
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
j : (gluedCover 𝒰).toGlueData.J
y : ↑↑(U (gluedCover 𝒰).toGlueData j).toPresheafedSpace
h : ↑(map 𝒰 i).val.base x = ↑(map 𝒰 j).val.base y
e : (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).pt ≅
  (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)
      (_ :
        forgetToTop.map pullback.fst ≫ forgetToTop.map (map 𝒰 i) =
          forgetToTop.map pullback.snd ≫ forgetToTop.map (map 𝒰 j))).pt :=
  IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j)))
    (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map 𝒰 i) (map 𝒰 j))
⊢ ↑(NatTrans.app (TopCat.pullbackCone (forgetToTop.map (map 𝒰 i)) (forgetToTop.map (map 𝒰 j))).π WalkingCospan.right)
      { val := (x, y), property := h } =
    y
[PROOFSTEP]
rfl
[GOAL]
X : Scheme
𝒰 : OpenCover X
x : ↑↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
⊢ IsIso (stalkMap (fromGlued 𝒰).val x)
[PROOFSTEP]
obtain ⟨i, x, rfl⟩ := 𝒰.gluedCover.ι_jointly_surjective x
[GOAL]
case intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
⊢ IsIso (stalkMap (fromGlued 𝒰).val (↑(GlueData.ι (gluedCover 𝒰) i).val.base x))
[PROOFSTEP]
have := PresheafedSpace.stalkMap.congr_hom _ _ (congr_arg LocallyRingedSpace.Hom.val <| 𝒰.ι_fromGlued i) x
[GOAL]
case intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
this :
  stalkMap (GlueData.ι (gluedCover 𝒰) i ≫ fromGlued 𝒰).val x =
    eqToHom
        (_ :
          stalk X.toPresheafedSpace (↑(GlueData.ι (gluedCover 𝒰) i ≫ fromGlued 𝒰).val.base x) =
            stalk X.toPresheafedSpace (↑(map 𝒰 i).val.base x)) ≫
      stalkMap (map 𝒰 i).val x
⊢ IsIso (stalkMap (fromGlued 𝒰).val (↑(GlueData.ι (gluedCover 𝒰) i).val.base x))
[PROOFSTEP]
erw [PresheafedSpace.stalkMap.comp] at this
[GOAL]
case intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
this :
  stalkMap (fromGlued 𝒰).val (↑(GlueData.ι (gluedCover 𝒰) i).val.base x) ≫
      stalkMap (GlueData.ι (gluedCover 𝒰) i).val x =
    eqToHom
        (_ :
          stalk X.toPresheafedSpace (↑(GlueData.ι (gluedCover 𝒰) i ≫ fromGlued 𝒰).val.base x) =
            stalk X.toPresheafedSpace (↑(map 𝒰 i).val.base x)) ≫
      stalkMap (map 𝒰 i).val x
⊢ IsIso (stalkMap (fromGlued 𝒰).val (↑(GlueData.ι (gluedCover 𝒰) i).val.base x))
[PROOFSTEP]
rw [← IsIso.eq_comp_inv] at this
[GOAL]
case intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
this :
  stalkMap (fromGlued 𝒰).val (↑(GlueData.ι (gluedCover 𝒰) i).val.base x) =
    (eqToHom
          (_ :
            stalk X.toPresheafedSpace (↑(GlueData.ι (gluedCover 𝒰) i ≫ fromGlued 𝒰).val.base x) =
              stalk X.toPresheafedSpace (↑(map 𝒰 i).val.base x)) ≫
        stalkMap (map 𝒰 i).val x) ≫
      inv (stalkMap (GlueData.ι (gluedCover 𝒰) i).val x)
⊢ IsIso (stalkMap (fromGlued 𝒰).val (↑(GlueData.ι (gluedCover 𝒰) i).val.base x))
[PROOFSTEP]
rw [this]
[GOAL]
case intro.intro
X : Scheme
𝒰 : OpenCover X
i : (gluedCover 𝒰).toGlueData.J
x : ↑↑(U (gluedCover 𝒰).toGlueData i).toPresheafedSpace
this :
  stalkMap (fromGlued 𝒰).val (↑(GlueData.ι (gluedCover 𝒰) i).val.base x) =
    (eqToHom
          (_ :
            stalk X.toPresheafedSpace (↑(GlueData.ι (gluedCover 𝒰) i ≫ fromGlued 𝒰).val.base x) =
              stalk X.toPresheafedSpace (↑(map 𝒰 i).val.base x)) ≫
        stalkMap (map 𝒰 i).val x) ≫
      inv (stalkMap (GlueData.ι (gluedCover 𝒰) i).val x)
⊢ IsIso
    ((eqToHom
          (_ :
            stalk X.toPresheafedSpace (↑(GlueData.ι (gluedCover 𝒰) i ≫ fromGlued 𝒰).val.base x) =
              stalk X.toPresheafedSpace (↑(map 𝒰 i).val.base x)) ≫
        stalkMap (map 𝒰 i).val x) ≫
      inv (stalkMap (GlueData.ι (gluedCover 𝒰) i).val x))
[PROOFSTEP]
infer_instance
[GOAL]
X : Scheme
𝒰 : OpenCover X
⊢ IsOpenMap ↑(fromGlued 𝒰).val.base
[PROOFSTEP]
intro U hU
[GOAL]
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : _root_.IsOpen U
⊢ _root_.IsOpen (↑(fromGlued 𝒰).val.base '' U)
[PROOFSTEP]
rw [isOpen_iff_forall_mem_open]
[GOAL]
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : _root_.IsOpen U
⊢ ∀ (x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace),
    x ∈ ↑(fromGlued 𝒰).val.base '' U → ∃ t, t ⊆ ↑(fromGlued 𝒰).val.base '' U ∧ _root_.IsOpen t ∧ x ∈ t
[PROOFSTEP]
intro x hx
[GOAL]
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : _root_.IsOpen U
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
⊢ ∃ t, t ⊆ ↑(fromGlued 𝒰).val.base '' U ∧ _root_.IsOpen t ∧ x ∈ t
[PROOFSTEP]
rw [𝒰.gluedCover.isOpen_iff] at hU
[GOAL]
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
⊢ ∃ t, t ⊆ ↑(fromGlued 𝒰).val.base '' U ∧ _root_.IsOpen t ∧ x ∈ t
[PROOFSTEP]
use 𝒰.fromGlued.val.base '' U ∩ Set.range (𝒰.map (𝒰.f x)).1.base
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
⊢ ↑(fromGlued 𝒰).val.base '' U ∩ Set.range ↑(map 𝒰 (f 𝒰 x)).val.base ⊆ ↑(fromGlued 𝒰).val.base '' U ∧
    _root_.IsOpen (↑(fromGlued 𝒰).val.base '' U ∩ Set.range ↑(map 𝒰 (f 𝒰 x)).val.base) ∧
      x ∈ ↑(fromGlued 𝒰).val.base '' U ∩ Set.range ↑(map 𝒰 (f 𝒰 x)).val.base
[PROOFSTEP]
use Set.inter_subset_left _ _
[GOAL]
case right
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
⊢ _root_.IsOpen (↑(fromGlued 𝒰).val.base '' U ∩ Set.range ↑(map 𝒰 (f 𝒰 x)).val.base) ∧
    x ∈ ↑(fromGlued 𝒰).val.base '' U ∩ Set.range ↑(map 𝒰 (f 𝒰 x)).val.base
[PROOFSTEP]
constructor
[GOAL]
case right.left
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
⊢ _root_.IsOpen (↑(fromGlued 𝒰).val.base '' U ∩ Set.range ↑(map 𝒰 (f 𝒰 x)).val.base)
[PROOFSTEP]
rw [← Set.image_preimage_eq_inter_range]
[GOAL]
case right.left
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
⊢ _root_.IsOpen (↑(map 𝒰 (f 𝒰 x)).val.base '' (↑(map 𝒰 (f 𝒰 x)).val.base ⁻¹' (↑(fromGlued 𝒰).val.base '' U)))
[PROOFSTEP]
apply (show IsOpenImmersion (𝒰.map (𝒰.f x)) from inferInstance).base_open.isOpenMap
[GOAL]
case right.left.a
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
⊢ _root_.IsOpen (↑(map 𝒰 (f 𝒰 x)).val.base ⁻¹' (↑(fromGlued 𝒰).val.base '' U))
[PROOFSTEP]
convert hU (𝒰.f x) using 1
[GOAL]
case h.e'_3.h
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
e_1✝ :
  (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace =
    (CategoryTheory.forget TopCat).obj ↑(CategoryTheory.GlueData.U (gluedCover 𝒰).toGlueData (f 𝒰 x)).toPresheafedSpace
⊢ ↑(map 𝒰 (f 𝒰 x)).val.base ⁻¹' (↑(fromGlued 𝒰).val.base '' U) = ↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base ⁻¹' U
[PROOFSTEP]
rw [← ι_fromGlued]
[GOAL]
case h.e'_3.h
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
e_1✝ :
  (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace =
    (CategoryTheory.forget TopCat).obj ↑(CategoryTheory.GlueData.U (gluedCover 𝒰).toGlueData (f 𝒰 x)).toPresheafedSpace
⊢ ↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x) ≫ fromGlued 𝒰).val.base ⁻¹' (↑(fromGlued 𝒰).val.base '' U) =
    ↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base ⁻¹' U
[PROOFSTEP]
erw [coe_comp]
[GOAL]
case h.e'_3.h
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
e_1✝ :
  (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace =
    (CategoryTheory.forget TopCat).obj ↑(CategoryTheory.GlueData.U (gluedCover 𝒰).toGlueData (f 𝒰 x)).toPresheafedSpace
⊢ ↑(fromGlued 𝒰).val.base ∘ ↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base ⁻¹' (↑(fromGlued 𝒰).val.base '' U) =
    ↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base ⁻¹' U
[PROOFSTEP]
rw [Set.preimage_comp]
[GOAL]
case h.e'_3.h
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
e_1✝ :
  (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace =
    (CategoryTheory.forget TopCat).obj ↑(CategoryTheory.GlueData.U (gluedCover 𝒰).toGlueData (f 𝒰 x)).toPresheafedSpace
⊢ ↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base ⁻¹' (↑(fromGlued 𝒰).val.base ⁻¹' (↑(fromGlued 𝒰).val.base '' U)) =
    ↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base ⁻¹' U
[PROOFSTEP]
congr! 1
[GOAL]
case h.e'_3.h.h.e'_4
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
e_1✝ :
  (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace =
    (CategoryTheory.forget TopCat).obj ↑(CategoryTheory.GlueData.U (gluedCover 𝒰).toGlueData (f 𝒰 x)).toPresheafedSpace
⊢ ↑(fromGlued 𝒰).val.base ⁻¹' (↑(fromGlued 𝒰).val.base '' U) = U
[PROOFSTEP]
refine' Set.preimage_image_eq _ 𝒰.fromGlued_injective
[GOAL]
case right.right
X : Scheme
𝒰 : OpenCover X
U :
  Set
    ((CategoryTheory.forget TopCat).obj
      ↑(GlueData.glued (gluedCover 𝒰)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)
hU : ∀ (i : (gluedCover 𝒰).toGlueData.J), _root_.IsOpen (↑(GlueData.ι (gluedCover 𝒰) i).val.base ⁻¹' U)
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
hx : x ∈ ↑(fromGlued 𝒰).val.base '' U
⊢ x ∈ ↑(fromGlued 𝒰).val.base '' U ∩ Set.range ↑(map 𝒰 (f 𝒰 x)).val.base
[PROOFSTEP]
exact ⟨hx, 𝒰.Covers x⟩
[GOAL]
X : Scheme
𝒰 : OpenCover X
⊢ Epi (fromGlued 𝒰).val.base
[PROOFSTEP]
rw [TopCat.epi_iff_surjective]
[GOAL]
X : Scheme
𝒰 : OpenCover X
⊢ Function.Surjective ↑(fromGlued 𝒰).val.base
[PROOFSTEP]
intro x
[GOAL]
X : Scheme
𝒰 : OpenCover X
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
⊢ ∃ a, ↑(fromGlued 𝒰).val.base a = x
[PROOFSTEP]
obtain ⟨y, h⟩ := 𝒰.Covers x
[GOAL]
case intro
X : Scheme
𝒰 : OpenCover X
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
y : (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace
h : ↑(map 𝒰 (f 𝒰 x)).val.base y = x
⊢ ∃ a, ↑(fromGlued 𝒰).val.base a = x
[PROOFSTEP]
use(𝒰.gluedCover.ι (𝒰.f x)).1.base y
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
y : (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace
h : ↑(map 𝒰 (f 𝒰 x)).val.base y = x
⊢ ↑(fromGlued 𝒰).val.base (↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base y) = x
[PROOFSTEP]
rw [← comp_apply]
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
y : (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace
h : ↑(map 𝒰 (f 𝒰 x)).val.base y = x
⊢ ↑((GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base ≫ (fromGlued 𝒰).val.base) y = x
[PROOFSTEP]
rw [← 𝒰.ι_fromGlued (𝒰.f x)] at h
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
x : (CategoryTheory.forget TopCat).obj ↑X.toPresheafedSpace
y : (CategoryTheory.forget TopCat).obj ↑(obj 𝒰 (f 𝒰 x)).toPresheafedSpace
h : ↑(GlueData.ι (gluedCover 𝒰) (f 𝒰 x) ≫ fromGlued 𝒰).val.base y = x
⊢ ↑((GlueData.ι (gluedCover 𝒰) (f 𝒰 x)).val.base ≫ (fromGlued 𝒰).val.base) y = x
[PROOFSTEP]
exact h
[GOAL]
X : Scheme
𝒰 : OpenCover X
F : Scheme ⥤ PresheafedSpace CommRingCat :=
  forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace ⋙ SheafedSpace.forgetToPresheafedSpace
⊢ IsIso (F.map (fromGlued 𝒰))
[PROOFSTEP]
change @IsIso (PresheafedSpace _) _ _ _ 𝒰.fromGlued.val
[GOAL]
X : Scheme
𝒰 : OpenCover X
F : Scheme ⥤ PresheafedSpace CommRingCat :=
  forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace ⋙ SheafedSpace.forgetToPresheafedSpace
⊢ IsIso (fromGlued 𝒰).val
[PROOFSTEP]
apply PresheafedSpace.IsOpenImmersion.to_iso
[GOAL]
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
⊢ X ⟶ Y
[PROOFSTEP]
refine' inv 𝒰.fromGlued ≫ _
[GOAL]
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
⊢ GlueData.glued (gluedCover 𝒰) ⟶ Y
[PROOFSTEP]
fapply Multicoequalizer.desc
[GOAL]
case k
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
⊢ (b : (diagram (gluedCover 𝒰).toGlueData).R) → MultispanIndex.right (diagram (gluedCover 𝒰).toGlueData) b ⟶ Y
case h
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
⊢ ∀ (a : (diagram (gluedCover 𝒰).toGlueData).L),
    MultispanIndex.fst (diagram (gluedCover 𝒰).toGlueData) a ≫
        ?k (MultispanIndex.fstFrom (diagram (gluedCover 𝒰).toGlueData) a) =
      MultispanIndex.snd (diagram (gluedCover 𝒰).toGlueData) a ≫
        ?k (MultispanIndex.sndFrom (diagram (gluedCover 𝒰).toGlueData) a)
[PROOFSTEP]
exact f
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
⊢ ∀ (a : (diagram (gluedCover 𝒰).toGlueData).L),
    MultispanIndex.fst (diagram (gluedCover 𝒰).toGlueData) a ≫
        f (MultispanIndex.fstFrom (diagram (gluedCover 𝒰).toGlueData) a) =
      MultispanIndex.snd (diagram (gluedCover 𝒰).toGlueData) a ≫
        f (MultispanIndex.sndFrom (diagram (gluedCover 𝒰).toGlueData) a)
[PROOFSTEP]
rintro ⟨i, j⟩
[GOAL]
case h.mk
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
i j : (gluedCover 𝒰).toGlueData.J
⊢ MultispanIndex.fst (diagram (gluedCover 𝒰).toGlueData) (i, j) ≫
      f (MultispanIndex.fstFrom (diagram (gluedCover 𝒰).toGlueData) (i, j)) =
    MultispanIndex.snd (diagram (gluedCover 𝒰).toGlueData) (i, j) ≫
      f (MultispanIndex.sndFrom (diagram (gluedCover 𝒰).toGlueData) (i, j))
[PROOFSTEP]
change pullback.fst ≫ f i = (_ ≫ _) ≫ f j
[GOAL]
case h.mk
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
i j : (gluedCover 𝒰).toGlueData.J
⊢ pullback.fst ≫ f i = (t (gluedCover 𝒰).toGlueData i j ≫ GlueData.f (gluedCover 𝒰).toGlueData j i) ≫ f j
[PROOFSTEP]
erw [pullbackSymmetry_hom_comp_fst]
[GOAL]
case h.mk
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
i j : (gluedCover 𝒰).toGlueData.J
⊢ pullback.fst ≫ f i = pullback.snd ≫ f j
[PROOFSTEP]
exact hf i j
[GOAL]
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
x : 𝒰.J
⊢ map 𝒰 x ≫ glueMorphisms 𝒰 f hf = f x
[PROOFSTEP]
rw [← ι_fromGlued, Category.assoc]
[GOAL]
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f : (x : 𝒰.J) → obj 𝒰 x ⟶ Y
hf : ∀ (x y : 𝒰.J), pullback.fst ≫ f x = pullback.snd ≫ f y
x : 𝒰.J
⊢ GlueData.ι (gluedCover 𝒰) x ≫ fromGlued 𝒰 ≫ glueMorphisms 𝒰 f hf = f x
[PROOFSTEP]
erw [IsIso.hom_inv_id_assoc, Multicoequalizer.π_desc]
[GOAL]
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f₁ f₂ : X ⟶ Y
h : ∀ (x : 𝒰.J), map 𝒰 x ≫ f₁ = map 𝒰 x ≫ f₂
⊢ f₁ = f₂
[PROOFSTEP]
rw [← cancel_epi 𝒰.fromGlued]
[GOAL]
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f₁ f₂ : X ⟶ Y
h : ∀ (x : 𝒰.J), map 𝒰 x ≫ f₁ = map 𝒰 x ≫ f₂
⊢ fromGlued 𝒰 ≫ f₁ = fromGlued 𝒰 ≫ f₂
[PROOFSTEP]
apply Multicoequalizer.hom_ext
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f₁ f₂ : X ⟶ Y
h : ∀ (x : 𝒰.J), map 𝒰 x ≫ f₁ = map 𝒰 x ≫ f₂
⊢ ∀ (b : (diagram (gluedCover 𝒰).toGlueData).R),
    Multicoequalizer.π (diagram (gluedCover 𝒰).toGlueData) b ≫ fromGlued 𝒰 ≫ f₁ =
      Multicoequalizer.π (diagram (gluedCover 𝒰).toGlueData) b ≫ fromGlued 𝒰 ≫ f₂
[PROOFSTEP]
intro x
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f₁ f₂ : X ⟶ Y
h : ∀ (x : 𝒰.J), map 𝒰 x ≫ f₁ = map 𝒰 x ≫ f₂
x : (diagram (gluedCover 𝒰).toGlueData).R
⊢ Multicoequalizer.π (diagram (gluedCover 𝒰).toGlueData) x ≫ fromGlued 𝒰 ≫ f₁ =
    Multicoequalizer.π (diagram (gluedCover 𝒰).toGlueData) x ≫ fromGlued 𝒰 ≫ f₂
[PROOFSTEP]
erw [Multicoequalizer.π_desc_assoc]
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f₁ f₂ : X ⟶ Y
h : ∀ (x : 𝒰.J), map 𝒰 x ≫ f₁ = map 𝒰 x ≫ f₂
x : (diagram (gluedCover 𝒰).toGlueData).R
⊢ map 𝒰 x ≫ f₁ = Multicoequalizer.π (diagram (gluedCover 𝒰).toGlueData) x ≫ fromGlued 𝒰 ≫ f₂
[PROOFSTEP]
erw [Multicoequalizer.π_desc_assoc]
[GOAL]
case h
X : Scheme
𝒰 : OpenCover X
Y : Scheme
f₁ f₂ : X ⟶ Y
h : ∀ (x : 𝒰.J), map 𝒰 x ≫ f₁ = map 𝒰 x ≫ f₂
x : (diagram (gluedCover 𝒰).toGlueData).R
⊢ map 𝒰 x ≫ f₁ = map 𝒰 x ≫ f₂
[PROOFSTEP]
exact h x