Module: Mathlib.AlgebraicGeometry.ZariskisMainTheorem

Goal state before:
pp: X S : Scheme
f : X ⟶ S
inst✝¹ : LocallyOfFiniteType f
inst✝ : IsSeparated f
x : ↥X
s : ↥S
h : f x = s
hx : Scheme.Hom.QuasiFiniteAt f x
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : f x ∈ ↑U
V : TopologicalSpace.Opens ↥X
hV : V ∈ X.affineOpens
hxV : x ∈ ↑V
hUV : V ≤ f ⁻¹ᵁ U
this✝³ : (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hUV)).FiniteType
algInst✝ : Algebra ↑Γ(S, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hUV)).toAlgebra
algebraizeInst✝ : Algebra.FiniteType ↑Γ(S, U) ↑Γ(X, V)
this✝² : (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal.LiesOver (IsAffineOpen.primeIdealOf hU ⟨f x, hxU⟩).asIdeal
this✝¹ : Algebra.QuasiFiniteAt (↑Γ(S, U)) (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal
R : Type u
w✝⁷ : CommRing R
w✝⁶ : Algebra (↑Γ(S, U)) R
w✝⁵ : Algebra.Etale (↑Γ(S, U)) R
P : Ideal R
w✝⁴ : P.IsPrime
w✝³ : P.LiesOver (IsAffineOpen.primeIdealOf hU ⟨f x, hxU⟩).asIdeal
e : R ⊗[↑Γ(S, U)] ↑Γ(X, V)
w✝² : IsIdempotentElem e
P' : Ideal (R ⊗[↑Γ(S, U)] ↑Γ(X, V))
w✝¹ : P'.IsPrime
w✝ : P'.LiesOver P
hP' : Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P' = (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal
heP' : e ∉ P'
left✝ : Module.Finite R (Localization.Away e)
this✝ : (algebraMap R (Localization.Away e)).Finite
φ : Γ(S, U) ⟶ CommRingCat.of R := CommRingCat.ofHom (algebraMap (↑Γ(S, U)) R)
hφ : (CommRingCat.Hom.hom φ).Etale
this : Etale (Spec.map φ)
e₁ : Spec (CommRingCat.of (R ⊗[↑Γ(S, U)] ↑Γ(X, V))) ≅ pullback (Spec.map (Scheme.Hom.appLE f U V hUV)) (Spec.map φ) :=
  (pullbackSpecIso (↑Γ(S, U)) R ↑Γ(X, V)).symm ≪≫
    pullbackSymmetry (Spec.map (CommRingCat.ofHom (algebraMap (↑Γ(S, U)) R)))
      (Spec.map (CommRingCat.ofHom (algebraMap ↑Γ(S, U) ↑Γ(X, V))))
⊢ ∃ U g,
    Etale g ∧ s ∈ Set.range ⇑g ∧ ∃ V W v, IsCompl V W ∧ IsFinite (V.ι ≫ pullback.snd f g) ∧ (pullback.fst f g) ↑v = x

Tactic:
have he₁ : e₁.hom ≫ pullback.fst _ _ =
      Spec.map (CommRingCat.ofHom Algebra.TensorProduct.includeRight.toRingHom) := by
    dsimp [e₁, RingHom.algebraMap_toAlgebra]
    rw [Category.assoc, pullbackSymmetry_hom_comp_fst]
    exact pullbackSpecIso_inv_snd ..

Goal state after:
X S : Scheme
f : X ⟶ S
inst✝¹ : LocallyOfFiniteType f
inst✝ : IsSeparated f
x : ↥X
s : ↥S
h : f x = s
hx : Scheme.Hom.QuasiFiniteAt f x
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : f x ∈ ↑U
V : TopologicalSpace.Opens ↥X
hV : V ∈ X.affineOpens
hxV : x ∈ ↑V
hUV : V ≤ f ⁻¹ᵁ U
this✝³ : (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hUV)).FiniteType
algInst✝ : Algebra ↑Γ(S, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hUV)).toAlgebra
algebraizeInst✝ : Algebra.FiniteType ↑Γ(S, U) ↑Γ(X, V)
this✝² : (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal.LiesOver (IsAffineOpen.primeIdealOf hU ⟨f x, hxU⟩).asIdeal
this✝¹ : Algebra.QuasiFiniteAt (↑Γ(S, U)) (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal
R : Type u
w✝⁷ : CommRing R
w✝⁶ : Algebra (↑Γ(S, U)) R
w✝⁵ : Algebra.Etale (↑Γ(S, U)) R
P : Ideal R
w✝⁴ : P.IsPrime
w✝³ : P.LiesOver (IsAffineOpen.primeIdealOf hU ⟨f x, hxU⟩).asIdeal
e : R ⊗[↑Γ(S, U)] ↑Γ(X, V)
w✝² : IsIdempotentElem e
P' : Ideal (R ⊗[↑Γ(S, U)] ↑Γ(X, V))
w✝¹ : P'.IsPrime
w✝ : P'.LiesOver P
hP' : Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P' = (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal
heP' : e ∉ P'
left✝ : Module.Finite R (Localization.Away e)
this✝ : (algebraMap R (Localization.Away e)).Finite
φ : Γ(S, U) ⟶ CommRingCat.of R := CommRingCat.ofHom (algebraMap (↑Γ(S, U)) R)
hφ : (CommRingCat.Hom.hom φ).Etale
this : Etale (Spec.map φ)
e₁ : Spec (CommRingCat.of (R ⊗[↑Γ(S, U)] ↑Γ(X, V))) ≅ pullback (Spec.map (Scheme.Hom.appLE f U V hUV)) (Spec.map φ) :=
  (pullbackSpecIso (↑Γ(S, U)) R ↑Γ(X, V)).symm ≪≫
    pullbackSymmetry (Spec.map (CommRingCat.ofHom (algebraMap (↑Γ(S, U)) R)))
      (Spec.map (CommRingCat.ofHom (algebraMap ↑Γ(S, U) ↑Γ(X, V))))
he₁ :
  e₁.hom ≫ pullback.fst (Spec.map (Scheme.Hom.appLE f U V hUV)) (Spec.map φ) =
    Spec.map (CommRingCat.ofHom Algebra.TensorProduct.includeRight.toRingHom)
⊢ ∃ U g,
    Etale g ∧ s ∈ Set.range ⇑g ∧ ∃ V W v, IsCompl V W ∧ IsFinite (V.ι ≫ pullback.snd f g) ∧ (pullback.fst f g) ↑v = x