Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: n : ℕ
IH :
  ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]
    [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p
R S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : WeaklyQuasiFiniteAt R p
f : MvPolynomial (Fin (n + 1)) R →ₐ[R] S
hf : f.Finite
f' : (MvPolynomial (Fin n) R)[X] →ₐ[R] S := f.comp ↑(MvPolynomial.finSuccEquiv R n).symm
this✝⁶ : Algebra (MvPolynomial (Fin n) R) S := (f'.comp C).toAlgebra
this✝⁵ : IsScalarTower R (MvPolynomial (Fin n) R) S
f'' : (MvPolynomial (Fin n) R)[X] →ₐ[MvPolynomial (Fin n) R] S := { toRingHom := f'.toRingHom, commutes' := ⋯ }
this✝⁴ : WeaklyQuasiFiniteAt (MvPolynomial (Fin n) R) p
this✝³ : ZariskisMainProperty (MvPolynomial (Fin n) R) p
r : S
hrp : r ∉ p
hr : IsIntegral (MvPolynomial (Fin n) R) r
m : S → ℕ
hm : ∀ (x : S), IsIntegral (MvPolynomial (Fin n) R) (r ^ m x * x)
s : Finset S
hs : adjoin R ↑s = ⊤
R' : Subalgebra R S := adjoin R ↑(Finset.image (⇑f ∘ MvPolynomial.X ∘ Fin.succ) Finset.univ ∪ r ^ s.sup m • s ∪ {r})
hrR' : r ∈ R'
this✝² : WeaklyQuasiFiniteAt R (Ideal.under (↥R') p)
φ : MvPolynomial (Fin n) R →ₐ[R] ↥R' := MvPolynomial.aeval fun i ↦ ⟨f (MvPolynomial.X i.succ), ⋯⟩
this✝¹ : Algebra (MvPolynomial (Fin n) R) ↥R' := φ.toAlgebra
this✝ : IsScalarTower (MvPolynomial (Fin n) R) (↥R') S
this :
  (∀ (a : Fin n), IsIntegral (MvPolynomial (Fin n) R) (f (MvPolynomial.X a.succ))) ∧
    ∀ a ∈ s, IsIntegral (MvPolynomial (Fin n) R) (r ^ s.sup m * a)
⊢ R'.val.range ≤ Subalgebra.map R'.val (Subalgebra.restrictScalars R (integralClosure (MvPolynomial (Fin n) R) ↥R'))

Tactic:
simp +contextual only [Subalgebra.range_val, Algebra.adjoin_le_iff, Subalgebra.coe_map,
            Subalgebra.coe_val, Set.subset_def, SetLike.mem_coe, Algebra.mem_adjoin_of_mem,
            Set.mem_image, Subtype.exists, exists_and_right, exists_eq_right, R']

Goal state after:
n : ℕ
IH :
  ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]
    [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p
R S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : WeaklyQuasiFiniteAt R p
f : MvPolynomial (Fin (n + 1)) R →ₐ[R] S
hf : f.Finite
f' : (MvPolynomial (Fin n) R)[X] →ₐ[R] S := f.comp ↑(MvPolynomial.finSuccEquiv R n).symm
this✝⁶ : Algebra (MvPolynomial (Fin n) R) S := (f'.comp C).toAlgebra
this✝⁵ : IsScalarTower R (MvPolynomial (Fin n) R) S
f'' : (MvPolynomial (Fin n) R)[X] →ₐ[MvPolynomial (Fin n) R] S := { toRingHom := f'.toRingHom, commutes' := ⋯ }
this✝⁴ : WeaklyQuasiFiniteAt (MvPolynomial (Fin n) R) p
this✝³ : ZariskisMainProperty (MvPolynomial (Fin n) R) p
r : S
hrp : r ∉ p
hr : IsIntegral (MvPolynomial (Fin n) R) r
m : S → ℕ
hm : ∀ (x : S), IsIntegral (MvPolynomial (Fin n) R) (r ^ m x * x)
s : Finset S
hs : adjoin R ↑s = ⊤
R' : Subalgebra R S := adjoin R ↑(Finset.image (⇑f ∘ MvPolynomial.X ∘ Fin.succ) Finset.univ ∪ r ^ s.sup m • s ∪ {r})
hrR' : r ∈ R'
this✝² : WeaklyQuasiFiniteAt R (Ideal.under (↥R') p)
φ : MvPolynomial (Fin n) R →ₐ[R] ↥R' := MvPolynomial.aeval fun i ↦ ⟨f (MvPolynomial.X i.succ), ⋯⟩
this✝¹ : Algebra (MvPolynomial (Fin n) R) ↥R' := φ.toAlgebra
this✝ : IsScalarTower (MvPolynomial (Fin n) R) (↥R') S
this :
  (∀ (a : Fin n), IsIntegral (MvPolynomial (Fin n) R) (f (MvPolynomial.X a.succ))) ∧
    ∀ a ∈ s, IsIntegral (MvPolynomial (Fin n) R) (r ^ s.sup m * a)
⊢ ∀ (x : S) (a : x ∈ Finset.image (⇑f ∘ MvPolynomial.X ∘ Fin.succ) Finset.univ ∪ r ^ s.sup m • s ∪ {r}),
    ∃ (h : True),
      ⟨x, ⋯⟩ ∈
        Subalgebra.restrictScalars R
          (integralClosure (MvPolynomial (Fin n) R)
            ↥(adjoin R ↑(Finset.image (⇑f ∘ MvPolynomial.X ∘ Fin.succ) Finset.univ ∪ r ^ s.sup m • s ∪ {r})))