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✝⁴ : Algebra (MvPolynomial (Fin n) R) S := (f'.comp C).toAlgebra
this✝³ : IsScalarTower R (MvPolynomial (Fin n) R) S
f'' : (MvPolynomial (Fin n) R)[X] →ₐ[MvPolynomial (Fin n) R] S := { toRingHom := f'.toRingHom, commutes' := ⋯ }
this✝² : WeaklyQuasiFiniteAt (MvPolynomial (Fin n) R) p
this✝¹ : 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✝ : WeaklyQuasiFiniteAt R (Ideal.under (↥R') p)
φ : MvPolynomial (Fin n) R →ₐ[R] ↥R' := MvPolynomial.aeval fun i ↦ ⟨f (MvPolynomial.X i.succ), ⋯⟩
this : Algebra (MvPolynomial (Fin n) R) ↥R' := φ.toAlgebra
⊢ algebraMap (MvPolynomial (Fin n) R) S = (algebraMap (↥R') S).comp (algebraMap (MvPolynomial (Fin n) R) ↥R')

Tactic:
ext

Goal state after:
case hC
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✝⁴ : Algebra (MvPolynomial (Fin n) R) S := (f'.comp C).toAlgebra
this✝³ : IsScalarTower R (MvPolynomial (Fin n) R) S
f'' : (MvPolynomial (Fin n) R)[X] →ₐ[MvPolynomial (Fin n) R] S := { toRingHom := f'.toRingHom, commutes' := ⋯ }
this✝² : WeaklyQuasiFiniteAt (MvPolynomial (Fin n) R) p
this✝¹ : 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✝ : WeaklyQuasiFiniteAt R (Ideal.under (↥R') p)
φ : MvPolynomial (Fin n) R →ₐ[R] ↥R' := MvPolynomial.aeval fun i ↦ ⟨f (MvPolynomial.X i.succ), ⋯⟩
this : Algebra (MvPolynomial (Fin n) R) ↥R' := φ.toAlgebra
x✝ : R
⊢ ((algebraMap (MvPolynomial (Fin n) R) S).comp MvPolynomial.C) x✝ =
    (((algebraMap (↥R') S).comp (algebraMap (MvPolynomial (Fin n) R) ↥R')).comp MvPolynomial.C) x✝

case hX
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✝⁴ : Algebra (MvPolynomial (Fin n) R) S := (f'.comp C).toAlgebra
this✝³ : IsScalarTower R (MvPolynomial (Fin n) R) S
f'' : (MvPolynomial (Fin n) R)[X] →ₐ[MvPolynomial (Fin n) R] S := { toRingHom := f'.toRingHom, commutes' := ⋯ }
this✝² : WeaklyQuasiFiniteAt (MvPolynomial (Fin n) R) p
this✝¹ : 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✝ : WeaklyQuasiFiniteAt R (Ideal.under (↥R') p)
φ : MvPolynomial (Fin n) R →ₐ[R] ↥R' := MvPolynomial.aeval fun i ↦ ⟨f (MvPolynomial.X i.succ), ⋯⟩
this : Algebra (MvPolynomial (Fin n) R) ↥R' := φ.toAlgebra
i✝ : Fin n
⊢ (algebraMap (MvPolynomial (Fin n) R) S) (MvPolynomial.X i✝) =
    ((algebraMap (↥R') S).comp (algebraMap (MvPolynomial (Fin n) R) ↥R')) (MvPolynomial.X i✝)