Theorem: Algebra.algebraMap_intTrace
File: Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean

Proof steps:
state_before: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
state_after: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
tactic: haveI : IsIntegralClosure B A (FractionRing B) :=
    IsIntegralClosure.of_isIntegrallyClosed _ _ _
state_before: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
state_after: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
tactic: haveI : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B) :=
    IsIntegralClosure.isLocalization _ (FractionRing A) _ _
state_before: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
state_after: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
tactic: haveI : FiniteDimensional (FractionRing A) (FractionRing B) :=
    Module.Finite_of_isLocalization A B _ _ A⁰
state_before: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
state_after: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
tactic: haveI := IsIntegralClosure.isFractionRing_of_finite_extension A K L B
state_before: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (algebraMap A K) ((intTrace A B) x) = (trace K L) ((algebraMap B L) x)
state_after: case a
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (FractionRing.algEquiv A K).symm ((algebraMap A K) ((intTrace A B) x)) =
    (FractionRing.algEquiv A K).symm ((trace K L) ((algebraMap B L) x))
tactic: apply (FractionRing.algEquiv A K).symm.injective
state_before: case a
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (FractionRing.algEquiv A K).symm ((algebraMap A K) ((intTrace A B) x)) =
    (FractionRing.algEquiv A K).symm ((trace K L) ((algebraMap B L) x))
state_after: case a
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) =
    (FractionRing.algEquiv A K).symm ((trace K L) ((FractionRing.algEquiv B L) ((algebraMap B (FractionRing B)) x)))
tactic: rw [AlgEquiv.commutes, Algebra.intTrace, Algebra.map_intTraceAux,
    ← AlgEquiv.commutes (FractionRing.algEquiv B L)]
state_before: case a
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) =
    (FractionRing.algEquiv A K).symm ((trace K L) ((FractionRing.algEquiv B L) ((algebraMap B (FractionRing B)) x)))
state_after: case a.he
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (algebraMap K L).comp ↑(FractionRing.algEquiv A K).toRingEquiv =
    (↑(FractionRing.algEquiv B L).toRingEquiv).comp (algebraMap (FractionRing A) (FractionRing B))
tactic: apply Algebra.trace_eq_of_equiv_equiv (FractionRing.algEquiv A K).toRingEquiv
    (FractionRing.algEquiv B L).toRingEquiv
state_before: case a.he
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ (algebraMap K L).comp ↑(FractionRing.algEquiv A K).toRingEquiv =
    (↑(FractionRing.algEquiv B L).toRingEquiv).comp (algebraMap (FractionRing A) (FractionRing B))
state_after: case a.he.h
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ ((algebraMap K L).comp ↑(FractionRing.algEquiv A K).toRingEquiv).comp (algebraMap A (FractionRing A)) =
    ((↑(FractionRing.algEquiv B L).toRingEquiv).comp (algebraMap (FractionRing A) (FractionRing B))).comp
      (algebraMap A (FractionRing A))
tactic: apply IsLocalization.ringHom_ext A⁰
state_before: case a.he.h
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.262054
Bₘ : Type ?u.262057
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsFractionRing B L
⊢ ((algebraMap K L).comp ↑(FractionRing.algEquiv A K).toRingEquiv).comp (algebraMap A (FractionRing A)) =
    ((↑(FractionRing.algEquiv B L).toRingEquiv).comp (algebraMap (FractionRing A) (FractionRing B))).comp
      (algebraMap A (FractionRing A))
state_after: no goals
tactic: simp only [AlgEquiv.toRingEquiv_eq_coe, ← AlgEquiv.coe_ringHom_commutes, RingHom.comp_assoc,
    AlgHom.comp_algebraMap_of_tower, ← IsScalarTower.algebraMap_eq, RingHom.comp_assoc]