Module: Mathlib.AlgebraicGeometry.Normalization

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝² : QuasiCompact f
inst✝¹ : QuasiSeparated f
T : Scheme
f₁ f₂ : normalization f ⟶ T
g : T ⟶ Y
inst✝ : IsAffineHom g
H₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂
hf₁ : f₁ ≫ g = fromNormalization f
hf₂ : f₂ ≫ g = fromNormalization f
U : (normalizationOpenCover f).I₀
this✝² : Algebra ↑Γ(Y, ↑U) ↑Γ(X, f ⁻¹ᵁ ↑U) := (CommRingCat.Hom.hom (app f ↑U)).toAlgebra
this✝¹ : IsAffineHom f₁
this✝ : IsAffineHom f₂
f₀ : X ⟶ T := toNormalization f ≫ f₁
hf₀ : f₀ = toNormalization f ≫ f₂
h₁ : f ⁻¹ᵁ ↑U ≤ f₀ ⁻¹ᵁ g ⁻¹ᵁ ↑U
h₁' : f ⁻¹ᵁ ↑U = toNormalization f ⁻¹ᵁ f₂ ⁻¹ᵁ g ⁻¹ᵁ ↑U
h₂ : fromNormalization f ⁻¹ᵁ ↑U = f₁ ⁻¹ᵁ g ⁻¹ᵁ ↑U
h₂' : fromNormalization f ⁻¹ᵁ ↑U = f₂ ⁻¹ᵁ g ⁻¹ᵁ ↑U
h₃ : f ⁻¹ᵁ ↑U = toNormalization f ⁻¹ᵁ fromNormalization f ⁻¹ᵁ ↑U
this :
  appLE (toNormalization f) (f₂ ⁻¹ᵁ g ⁻¹ᵁ ↑U) (f ⁻¹ᵁ ↑U) ⋯ =
    (normalization f).presheaf.map (eqToHom h₂').op ≫
      app (toNormalization f) (fromNormalization f ⁻¹ᵁ ↑U) ≫ X.presheaf.map (eqToHom h₃).op
⊢ Spec.map (𝟙 Γ(X, f ⁻¹ᵁ ↑U)) ≫
      Spec.map (CommRingCat.ofHom ↑(integralClosure ↑Γ(Y, ↑U) ↑Γ(X, f ⁻¹ᵁ ↑U)).val) ≫
        Spec.map
            (ΓSpecIso
                (CommRingCat.of
                  ↥(integralClosure ↑Γ(Y, (toOpensFunctor Y).obj ⟨↑U, ⋯⟩)
                      ↑Γ(X, f ⁻¹ᵁ (toOpensFunctor Y).obj ⟨↑U, ⋯⟩)))).hom ≫
          Spec.map
              (ΓSpecIso
                  (CommRingCat.of
                    ↥(integralClosure ↑Γ(Y, (toOpensFunctor Y).obj ⟨↑U, ⋯⟩)
                        ↑Γ(X, f ⁻¹ᵁ (toOpensFunctor Y).obj ⟨↑U, ⋯⟩)))).inv ≫
            (normalizationOpenCover f).f ⟨↑U, ⋯⟩ ≫ f₂ =
    Spec.map (CommRingCat.ofHom ↑(integralClosure ↑Γ(Y, ↑U) ↑Γ(X, f ⁻¹ᵁ ↑U)).val) ≫ (normalizationOpenCover f).f U ≫ f₂

Tactic:
simp [← Spec.map_comp_assoc, -Spec.map_comp]

Goal state after:
X Y : Scheme
f : X ⟶ Y
inst✝² : QuasiCompact f
inst✝¹ : QuasiSeparated f
T : Scheme
f₁ f₂ : normalization f ⟶ T
g : T ⟶ Y
inst✝ : IsAffineHom g
H₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂
hf₁ : f₁ ≫ g = fromNormalization f
hf₂ : f₂ ≫ g = fromNormalization f
U : (normalizationOpenCover f).I₀
this✝² : Algebra ↑Γ(Y, ↑U) ↑Γ(X, f ⁻¹ᵁ ↑U) := (CommRingCat.Hom.hom (app f ↑U)).toAlgebra
this✝¹ : IsAffineHom f₁
this✝ : IsAffineHom f₂
f₀ : X ⟶ T := toNormalization f ≫ f₁
hf₀ : f₀ = toNormalization f ≫ f₂
h₁ : f ⁻¹ᵁ ↑U ≤ f₀ ⁻¹ᵁ g ⁻¹ᵁ ↑U
h₁' : f ⁻¹ᵁ ↑U = toNormalization f ⁻¹ᵁ f₂ ⁻¹ᵁ g ⁻¹ᵁ ↑U
h₂ : fromNormalization f ⁻¹ᵁ ↑U = f₁ ⁻¹ᵁ g ⁻¹ᵁ ↑U
h₂' : fromNormalization f ⁻¹ᵁ ↑U = f₂ ⁻¹ᵁ g ⁻¹ᵁ ↑U
h₃ : f ⁻¹ᵁ ↑U = toNormalization f ⁻¹ᵁ fromNormalization f ⁻¹ᵁ ↑U
this :
  appLE (toNormalization f) (f₂ ⁻¹ᵁ g ⁻¹ᵁ ↑U) (f ⁻¹ᵁ ↑U) ⋯ =
    (normalization f).presheaf.map (eqToHom h₂').op ≫
      app (toNormalization f) (fromNormalization f ⁻¹ᵁ ↑U) ≫ X.presheaf.map (eqToHom h₃).op
⊢ Spec.map (CommRingCat.ofHom ↑(integralClosure ↑Γ(Y, ↑U) ↑Γ(X, f ⁻¹ᵁ ↑U)).val ≫ 𝟙 Γ(X, f ⁻¹ᵁ ↑U)) ≫
      𝟙
          (Spec
            (CommRingCat.of
              ↥(integralClosure ↑Γ(Y, (toOpensFunctor Y).obj ⟨↑U, ⋯⟩) ↑Γ(X, f ⁻¹ᵁ (toOpensFunctor Y).obj ⟨↑U, ⋯⟩)))) ≫
        (normalizationOpenCover f).f ⟨↑U, ⋯⟩ ≫ f₂ =
    Spec.map (CommRingCat.ofHom ↑(integralClosure ↑Γ(Y, ↑U) ↑Γ(X, f ⁻¹ᵁ ↑U)).val) ≫ (normalizationOpenCover f).f U ≫ f₂