Module: Mathlib.AlgebraicGeometry.Morphisms.Flat

Goal state before:
pp: case intro.h₁
X Y S T : Scheme
f : T ⟶ S
g : Y ⟶ X
iX : X ⟶ S
iY : Y ⟶ T
H : IsPullback g iY iX f
US : S.Opens
UT : T.Opens
UX : X.Opens
hUST : UT ≤ f ⁻¹ᵁ US
hUSX : UX ≤ iX ⁻¹ᵁ US
UY : Y.Opens
hUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT
ι : Type u_1
inst✝ : Finite ι
VX : ι → X.Opens
hVU : iSup VX = UX
hV✝ : ∀ (i : ι), Mono (pushoutSection H hUST ⋯ ⋯)
hT : (CommRingCat.Hom.hom (Scheme.Hom.appLE f US UT hUST)).Flat
this✝ : ∀ (i : ι), VX i ≤ iX ⁻¹ᵁ US
algInst✝¹ : Algebra ↑Γ(S, US) ↑Γ(X, UX) := (CommRingCat.Hom.hom (Scheme.Hom.appLE iX US UX hUSX)).toAlgebra
algInst✝ : Algebra ↑Γ(S, US) ↑Γ(T, UT) := (CommRingCat.Hom.hom (Scheme.Hom.appLE f US UT hUST)).toAlgebra
algebraizeInst✝ : Module.Flat ↑Γ(S, US) ↑Γ(T, UT)
this : (i : ι) → Algebra ↑Γ(S, US) ↑Γ(X, VX i) :=
  fun i ↦ (CommRingCat.Hom.hom (Scheme.Hom.appLE iX US (VX i) ⋯)).toAlgebra
ψY : ↑Γ(Y, UY) →+* (i : ι) → ↑Γ(Y, g ⁻¹ᵁ VX i ⊓ iY ⁻¹ᵁ UT) :=
  RingHom.pi fun i ↦ CommRingCat.Hom.hom (Y.presheaf.map (homOfLE ⋯).op)
ψ : ↑Γ(X, UX) →ₐ[↑Γ(S, US)] (i : ι) → ↑Γ(X, VX i) :=
  AlgHom.pi fun i ↦ { toRingHom := CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op), commutes' := ⋯ }
hψ : Function.Injective ⇑ψ
hψ' : Function.Injective ⇑(Algebra.TensorProduct.map (AlgHom.id ↑Γ(T, UT) ↑Γ(T, UT)) ψ)
hV : ∀ (i : ι), Function.Injective ⇑(ConcreteCategory.hom (pushoutSection H hUST ⋯ ⋯))
val✝ : Fintype ι
φ : ↑Γ(T, UT) ⊗[↑Γ(S, US)] ((i : ι) → ↑Γ(X, VX i)) →+* (i : ι) → ↑Γ(Y, g ⁻¹ᵁ VX i ⊓ iY ⁻¹ᵁ UT) :=
  (RingHom.pi fun i ↦
        (CommRingCat.Hom.hom (pushoutSection H hUST ⋯ ⋯)).comp
          ((CommRingCat.Hom.hom ⋯.isoPushout.hom).comp
            (Pi.evalRingHom (fun i ↦ ↑Γ(T, UT) ⊗[↑Γ(S, US)] ↑Γ(X, VX i)) i))).comp
    (Algebra.TensorProduct.piRight ↑Γ(S, US) ↑Γ(S, US) ↑Γ(T, UT) fun i ↦ ↑Γ(X, VX i)).toRingEquiv.toRingHom
hφ : Function.Injective ⇑φ
e : pushout (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST) ≅
  CommRingCat.of (↑Γ(T, UT) ⊗[↑Γ(S, US)] ↑Γ(X, UX)) :=
  ⋯.isoPushout.symm
H₁ :
  (CommRingCat.Hom.hom e.inv).comp Algebra.TensorProduct.includeLeftRingHom =
    CommRingCat.Hom.hom (pushout.inr (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST))
H₂ :
  ∀ (x : ↑Γ(T, UT)) (j : ι),
    φ (x ⊗ₜ[↑Γ(S, US)] 1) j =
      (ConcreteCategory.hom (pushoutSection H hUST ⋯ ⋯))
        ((ConcreteCategory.hom (pushout.inr (Scheme.Hom.appLE iX US (VX j) ⋯) (Scheme.Hom.appLE f US UT hUST))) x)
⊢ ψY.comp
      ((CommRingCat.Hom.hom (pushoutSection H hUST hUSX hUY)).comp
        (CommRingCat.Hom.hom (pushout.inr (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST)))) =
    φ.comp ((Algebra.TensorProduct.map (AlgHom.id ↑Γ(T, UT) ↑Γ(T, UT)) ψ).comp Algebra.TensorProduct.includeLeftRingHom)

Tactic:
ext x j

Goal state after:
case intro.h₁
X Y S T : Scheme
f : T ⟶ S
g : Y ⟶ X
iX : X ⟶ S
iY : Y ⟶ T
H : IsPullback g iY iX f
US : S.Opens
UT : T.Opens
UX : X.Opens
hUST : UT ≤ f ⁻¹ᵁ US
hUSX : UX ≤ iX ⁻¹ᵁ US
UY : Y.Opens
hUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT
ι : Type u_1
inst✝ : Finite ι
VX : ι → X.Opens
hVU : iSup VX = UX
hV✝ : ∀ (i : ι), Mono (pushoutSection H hUST ⋯ ⋯)
hT : (CommRingCat.Hom.hom (Scheme.Hom.appLE f US UT hUST)).Flat
this✝ : ∀ (i : ι), VX i ≤ iX ⁻¹ᵁ US
algInst✝¹ : Algebra ↑Γ(S, US) ↑Γ(X, UX) := (CommRingCat.Hom.hom (Scheme.Hom.appLE iX US UX hUSX)).toAlgebra
algInst✝ : Algebra ↑Γ(S, US) ↑Γ(T, UT) := (CommRingCat.Hom.hom (Scheme.Hom.appLE f US UT hUST)).toAlgebra
algebraizeInst✝ : Module.Flat ↑Γ(S, US) ↑Γ(T, UT)
this : (i : ι) → Algebra ↑Γ(S, US) ↑Γ(X, VX i) :=
  fun i ↦ (CommRingCat.Hom.hom (Scheme.Hom.appLE iX US (VX i) ⋯)).toAlgebra
ψY : ↑Γ(Y, UY) →+* (i : ι) → ↑Γ(Y, g ⁻¹ᵁ VX i ⊓ iY ⁻¹ᵁ UT) :=
  RingHom.pi fun i ↦ CommRingCat.Hom.hom (Y.presheaf.map (homOfLE ⋯).op)
ψ : ↑Γ(X, UX) →ₐ[↑Γ(S, US)] (i : ι) → ↑Γ(X, VX i) :=
  AlgHom.pi fun i ↦ { toRingHom := CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op), commutes' := ⋯ }
hψ : Function.Injective ⇑ψ
hψ' : Function.Injective ⇑(Algebra.TensorProduct.map (AlgHom.id ↑Γ(T, UT) ↑Γ(T, UT)) ψ)
hV : ∀ (i : ι), Function.Injective ⇑(ConcreteCategory.hom (pushoutSection H hUST ⋯ ⋯))
val✝ : Fintype ι
φ : ↑Γ(T, UT) ⊗[↑Γ(S, US)] ((i : ι) → ↑Γ(X, VX i)) →+* (i : ι) → ↑Γ(Y, g ⁻¹ᵁ VX i ⊓ iY ⁻¹ᵁ UT) :=
  (RingHom.pi fun i ↦
        (CommRingCat.Hom.hom (pushoutSection H hUST ⋯ ⋯)).comp
          ((CommRingCat.Hom.hom ⋯.isoPushout.hom).comp
            (Pi.evalRingHom (fun i ↦ ↑Γ(T, UT) ⊗[↑Γ(S, US)] ↑Γ(X, VX i)) i))).comp
    (Algebra.TensorProduct.piRight ↑Γ(S, US) ↑Γ(S, US) ↑Γ(T, UT) fun i ↦ ↑Γ(X, VX i)).toRingEquiv.toRingHom
hφ : Function.Injective ⇑φ
e : pushout (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST) ≅
  CommRingCat.of (↑Γ(T, UT) ⊗[↑Γ(S, US)] ↑Γ(X, UX)) :=
  ⋯.isoPushout.symm
H₁ :
  (CommRingCat.Hom.hom e.inv).comp Algebra.TensorProduct.includeLeftRingHom =
    CommRingCat.Hom.hom (pushout.inr (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST))
H₂ :
  ∀ (x : ↑Γ(T, UT)) (j : ι),
    φ (x ⊗ₜ[↑Γ(S, US)] 1) j =
      (ConcreteCategory.hom (pushoutSection H hUST ⋯ ⋯))
        ((ConcreteCategory.hom (pushout.inr (Scheme.Hom.appLE iX US (VX j) ⋯) (Scheme.Hom.appLE f US UT hUST))) x)
x : ↑Γ(T, UT)
j : ι
⊢ (ψY.comp
        ((CommRingCat.Hom.hom (pushoutSection H hUST hUSX hUY)).comp
          (CommRingCat.Hom.hom (pushout.inr (Scheme.Hom.appLE iX US UX hUSX) (Scheme.Hom.appLE f US UT hUST)))))
      x j =
    (φ.comp
        ((Algebra.TensorProduct.map (AlgHom.id ↑Γ(T, UT) ↑Γ(T, UT)) ψ).comp Algebra.TensorProduct.includeLeftRingHom))
      x j