Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: case neg
A : Type u_1
B : Type u_6
inst✝²⁵ : CommRing A
inst✝²⁴ : CommRing B
inst✝²³ : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²² : CommRing Aₘ
inst✝²¹ : CommRing Bₘ
inst✝²⁰ : Algebra Aₘ Bₘ
inst✝¹⁹ : Algebra A Aₘ
inst✝¹⁸ : Algebra B Bₘ
inst✝¹⁷ : Algebra A Bₘ
inst✝¹⁶ : IsScalarTower A Aₘ Bₘ
inst✝¹⁵ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁴ : IsLocalization M Aₘ
inst✝¹³ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsDomain B
inst✝⁹ : IsIntegrallyClosed B
inst✝⁸ : Algebra.IsIntegral A B
inst✝⁷ : IsTorsionFree A B
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : IsTorsionFree Aₘ Bₘ
inst✝¹ : Algebra.IsIntegral Aₘ Bₘ
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝⁵ : Algebra Aₘ K := f.toAlgebra
this✝⁴ : IsScalarTower A Aₘ K
this✝³ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝² : Algebra Bₘ L := g.toAlgebra
this✝¹ : IsScalarTower B Bₘ L
this✝ : Algebra Aₘ L := ((algebraMap K L).comp f).toAlgebra
this : IsScalarTower Aₘ K L
⊢ (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)

Tactic:
have : IsScalarTower Aₘ Bₘ L := by
    apply IsScalarTower.of_algebraMap_eq'
    apply IsLocalization.ringHom_ext M
    rw [RingHom.algebraMap_toAlgebra, RingHom.algebraMap_toAlgebra (R := Bₘ), RingHom.comp_assoc,
      RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq, IsScalarTower.algebraMap_eq A B Bₘ,
      IsLocalization.map_comp, RingHom.comp_id, ← RingHom.comp_assoc, IsLocalization.map_comp,
      RingHom.comp_id, ← IsScalarTower.algebraMap_eq, ← IsScalarTower.algebraMap_eq]

Goal state after:
case neg
A : Type u_1
B : Type u_6
inst✝²⁵ : CommRing A
inst✝²⁴ : CommRing B
inst✝²³ : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²² : CommRing Aₘ
inst✝²¹ : CommRing Bₘ
inst✝²⁰ : Algebra Aₘ Bₘ
inst✝¹⁹ : Algebra A Aₘ
inst✝¹⁸ : Algebra B Bₘ
inst✝¹⁷ : Algebra A Bₘ
inst✝¹⁶ : IsScalarTower A Aₘ Bₘ
inst✝¹⁵ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁴ : IsLocalization M Aₘ
inst✝¹³ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsDomain B
inst✝⁹ : IsIntegrallyClosed B
inst✝⁸ : Algebra.IsIntegral A B
inst✝⁷ : IsTorsionFree A B
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : IsTorsionFree Aₘ Bₘ
inst✝¹ : Algebra.IsIntegral Aₘ Bₘ
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝⁶ : Algebra Aₘ K := f.toAlgebra
this✝⁵ : IsScalarTower A Aₘ K
this✝⁴ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝³ : Algebra Bₘ L := g.toAlgebra
this✝² : IsScalarTower B Bₘ L
this✝¹ : Algebra Aₘ L := ((algebraMap K L).comp f).toAlgebra
this✝ : IsScalarTower Aₘ K L
this : IsScalarTower Aₘ Bₘ L
⊢ (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)