Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

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

Tactic:
← IsScalarTower.algebraMap_apply

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