Module: Mathlib.RingTheory.RingHom.StandardSmooth

Goal state before:
pp: n : ℕ
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S
this✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite
ι σ : Type
w✝¹ : _root_.Finite σ
w✝ : _root_.Finite ι
P : Algebra.SubmersivePresentation R S ι σ
e✝ : P.dimension = n
e₀ : σ ⊕ Fin n ≃ ι :=
  ((Equiv.ofInjective P.map ⋯).sumCongr (Finite.equivFinOfCardEq ⋯).symm).trans (Equiv.Set.sumCompl (Set.range P.map))
e : MvPolynomial σ (MvPolynomial (Fin n) R) ≃ₐ[R] P.Ring :=
  (MvPolynomial.sumAlgEquiv R σ (Fin n)).symm.trans (MvPolynomial.renameEquiv R e₀)
φ : MvPolynomial (Fin n) R →ₐ[R] P.Ring :=
  (↑e).comp (IsScalarTower.toAlgHom R (MvPolynomial (Fin n) R) (MvPolynomial σ (MvPolynomial (Fin n) R)))
algInst✝¹ : Algebra (MvPolynomial (Fin n) R) P.Ring := φ.toAlgebra
algInst✝ : Algebra (MvPolynomial (Fin n) R) S := ((algebraMap P.Ring S).comp φ.toRingHom).toAlgebra
scalarTowerInst✝ : IsScalarTower (MvPolynomial (Fin n) R) P.Ring S :=
  IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap (MvPolynomial (Fin n) R) S))
this✝ : IsScalarTower R (MvPolynomial (Fin n) R) P.Ring
this : IsScalarTower R (MvPolynomial (Fin n) R) S
⊢ (MvPolynomial.aeval fun x ↦ (algebraMap P.Ring S) (e (MvPolynomial.X x))).toRingHom =
    (algebraMap P.Ring S).comp e.toRingEquiv.toRingHom

Tactic:
ext

Goal state after:
case hC.hC
n : ℕ
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S
this✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite
ι σ : Type
w✝¹ : _root_.Finite σ
w✝ : _root_.Finite ι
P : Algebra.SubmersivePresentation R S ι σ
e✝ : P.dimension = n
e₀ : σ ⊕ Fin n ≃ ι :=
  ((Equiv.ofInjective P.map ⋯).sumCongr (Finite.equivFinOfCardEq ⋯).symm).trans (Equiv.Set.sumCompl (Set.range P.map))
e : MvPolynomial σ (MvPolynomial (Fin n) R) ≃ₐ[R] P.Ring :=
  (MvPolynomial.sumAlgEquiv R σ (Fin n)).symm.trans (MvPolynomial.renameEquiv R e₀)
φ : MvPolynomial (Fin n) R →ₐ[R] P.Ring :=
  (↑e).comp (IsScalarTower.toAlgHom R (MvPolynomial (Fin n) R) (MvPolynomial σ (MvPolynomial (Fin n) R)))
algInst✝¹ : Algebra (MvPolynomial (Fin n) R) P.Ring := φ.toAlgebra
algInst✝ : Algebra (MvPolynomial (Fin n) R) S := ((algebraMap P.Ring S).comp φ.toRingHom).toAlgebra
scalarTowerInst✝ : IsScalarTower (MvPolynomial (Fin n) R) P.Ring S :=
  IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap (MvPolynomial (Fin n) R) S))
this✝ : IsScalarTower R (MvPolynomial (Fin n) R) P.Ring
this : IsScalarTower R (MvPolynomial (Fin n) R) S
x✝ : R
⊢ (((MvPolynomial.aeval fun x ↦ (algebraMap P.Ring S) (e (MvPolynomial.X x))).comp MvPolynomial.C).comp MvPolynomial.C)
      x✝ =
    ((((algebraMap P.Ring S).comp e.toRingEquiv.toRingHom).comp MvPolynomial.C).comp MvPolynomial.C) x✝

case hC.hX
n : ℕ
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S
this✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite
ι σ : Type
w✝¹ : _root_.Finite σ
w✝ : _root_.Finite ι
P : Algebra.SubmersivePresentation R S ι σ
e✝ : P.dimension = n
e₀ : σ ⊕ Fin n ≃ ι :=
  ((Equiv.ofInjective P.map ⋯).sumCongr (Finite.equivFinOfCardEq ⋯).symm).trans (Equiv.Set.sumCompl (Set.range P.map))
e : MvPolynomial σ (MvPolynomial (Fin n) R) ≃ₐ[R] P.Ring :=
  (MvPolynomial.sumAlgEquiv R σ (Fin n)).symm.trans (MvPolynomial.renameEquiv R e₀)
φ : MvPolynomial (Fin n) R →ₐ[R] P.Ring :=
  (↑e).comp (IsScalarTower.toAlgHom R (MvPolynomial (Fin n) R) (MvPolynomial σ (MvPolynomial (Fin n) R)))
algInst✝¹ : Algebra (MvPolynomial (Fin n) R) P.Ring := φ.toAlgebra
algInst✝ : Algebra (MvPolynomial (Fin n) R) S := ((algebraMap P.Ring S).comp φ.toRingHom).toAlgebra
scalarTowerInst✝ : IsScalarTower (MvPolynomial (Fin n) R) P.Ring S :=
  IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap (MvPolynomial (Fin n) R) S))
this✝ : IsScalarTower R (MvPolynomial (Fin n) R) P.Ring
this : IsScalarTower R (MvPolynomial (Fin n) R) S
i✝ : Fin n
⊢ ((MvPolynomial.aeval fun x ↦ (algebraMap P.Ring S) (e (MvPolynomial.X x))).comp MvPolynomial.C) (MvPolynomial.X i✝) =
    (((algebraMap P.Ring S).comp e.toRingEquiv.toRingHom).comp MvPolynomial.C) (MvPolynomial.X i✝)

case hX
n : ℕ
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S
this✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite
ι σ : Type
w✝¹ : _root_.Finite σ
w✝ : _root_.Finite ι
P : Algebra.SubmersivePresentation R S ι σ
e✝ : P.dimension = n
e₀ : σ ⊕ Fin n ≃ ι :=
  ((Equiv.ofInjective P.map ⋯).sumCongr (Finite.equivFinOfCardEq ⋯).symm).trans (Equiv.Set.sumCompl (Set.range P.map))
e : MvPolynomial σ (MvPolynomial (Fin n) R) ≃ₐ[R] P.Ring :=
  (MvPolynomial.sumAlgEquiv R σ (Fin n)).symm.trans (MvPolynomial.renameEquiv R e₀)
φ : MvPolynomial (Fin n) R →ₐ[R] P.Ring :=
  (↑e).comp (IsScalarTower.toAlgHom R (MvPolynomial (Fin n) R) (MvPolynomial σ (MvPolynomial (Fin n) R)))
algInst✝¹ : Algebra (MvPolynomial (Fin n) R) P.Ring := φ.toAlgebra
algInst✝ : Algebra (MvPolynomial (Fin n) R) S := ((algebraMap P.Ring S).comp φ.toRingHom).toAlgebra
scalarTowerInst✝ : IsScalarTower (MvPolynomial (Fin n) R) P.Ring S :=
  IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap (MvPolynomial (Fin n) R) S))
this✝ : IsScalarTower R (MvPolynomial (Fin n) R) P.Ring
this : IsScalarTower R (MvPolynomial (Fin n) R) S
i✝ : σ
⊢ (MvPolynomial.aeval fun x ↦ (algebraMap P.Ring S) (e (MvPolynomial.X x))).toRingHom (MvPolynomial.X i✝) =
    ((algebraMap P.Ring S).comp e.toRingEquiv.toRingHom) (MvPolynomial.X i✝)