ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K
inst✝³⁴ : Field L
inst✝³³ : Algebra A K
inst✝³² : IsFractionRing A K
inst✝³¹ : Algebra B L
inst✝³⁰ : Algebra K L
inst✝²⁹ : Algebra A L
inst✝²⁸ : IsScalarTower A B L
inst✝²⁷ : IsScalarTower A K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : FiniteDimensional K L
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : 0 ∉ M
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K
inst✝³⁴ : Field L
inst✝³³ : Algebra A K
inst✝³² : IsFractionRing A K
inst✝³¹ : Algebra B L
inst✝³⁰ : Algebra K L
inst✝²⁹ : Algebra A L
inst✝²⁸ : IsScalarTower A B L
inst✝²⁷ : IsScalarTower A K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : FiniteDimensional K L
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this : IsIntegralClosure B A L
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝ : IsIntegralClosure B A L
this : IsLocalization (algebraMapSubmonoid B A⁰) L
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝¹ : IsIntegralClosure B A L
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this : Algebra Aₘ K := f.toAlgebra
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝² : IsIntegralClosure B A L
this✝¹ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝ : Algebra Aₘ K := f.toAlgebra
this : IsScalarTower A Aₘ K
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝³ : IsIntegralClosure B A L
this✝² : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝¹ : Algebra Aₘ K := f.toAlgebra
this✝ : IsScalarTower A Aₘ K
this : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝³ : IsIntegralClosure B A L
this✝² : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝¹ : Algebra Aₘ K := f.toAlgebra
this✝ : IsScalarTower A Aₘ K
this : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝⁴ : IsIntegralClosure B A L
this✝³ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝² : Algebra Aₘ K := f.toAlgebra
this✝¹ : IsScalarTower A Aₘ K
this✝ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this : Algebra Bₘ L := g.toAlgebra
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝⁵ : IsIntegralClosure B A L
this✝⁴ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝³ : Algebra Aₘ K := f.toAlgebra
this✝² : IsScalarTower A Aₘ K
this✝¹ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝ : Algebra Bₘ L := g.toAlgebra
this : IsScalarTower B Bₘ L
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝⁶ : IsIntegralClosure B A L
this✝⁵ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝⁴ : Algebra Aₘ K := f.toAlgebra
this✝³ : IsScalarTower A Aₘ K
this✝² : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝¹ : Algebra Bₘ L := g.toAlgebra
this✝ : IsScalarTower B Bₘ L
this : Algebra Aₘ L := ((algebraMap K L).comp f).toAlgebra
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝⁷ : IsIntegralClosure B A L
this✝⁶ : IsLocalization (algebraMapSubmonoid B A⁰) L
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
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝⁸ : IsIntegralClosure B A L
this✝⁷ : IsLocalization (algebraMapSubmonoid B A⁰) L
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
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁸ : CommRing A
inst✝³⁷ : CommRing B
inst✝³⁶ : Algebra A B
inst✝³⁵ : Field K✝
inst✝³⁴ : Field L✝
inst✝³³ : Algebra A K✝
inst✝³² : IsFractionRing A K✝
inst✝³¹ : Algebra B L✝
inst✝³⁰ : Algebra K✝ L✝
inst✝²⁹ : Algebra A L✝
inst✝²⁸ : IsScalarTower A B L✝
inst✝²⁷ : IsScalarTower A K✝ L✝
inst✝²⁶ : IsIntegralClosure B A L✝
inst✝²⁵ : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²⁴ : CommRing Aₘ
inst✝²³ : CommRing Bₘ
inst✝²² : Algebra Aₘ Bₘ
inst✝²¹ : Algebra A Aₘ
inst✝²⁰ : Algebra B Bₘ
inst✝¹⁹ : Algebra A Bₘ
inst✝¹⁸ : IsScalarTower A Aₘ Bₘ
inst✝¹⁷ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁶ : IsLocalization M Aₘ
inst✝¹⁵ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹⁴ : IsIntegrallyClosed A
inst✝¹³ : IsDomain A
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsIntegrallyClosed B
inst✝⁹ : Module.Finite A B
inst✝⁸ : NoZeroSMulDivisors A B
inst✝⁷ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : NoZeroSMulDivisors Aₘ Bₘ
inst✝¹ : Module.Finite Aₘ Bₘ
inst✝ : Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝⁹ : IsIntegralClosure B A L
this✝⁸ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝⁷ : Algebra Aₘ K := f.toAlgebra
this✝⁶ : IsScalarTower A Aₘ K
this✝⁵ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝⁴ : Algebra Bₘ L := g.toAlgebra
this✝³ : IsScalarTower B Bₘ L
this✝² : Algebra Aₘ L := ((algebraMap K L).comp f).toAlgebra
this✝¹ : IsScalarTower Aₘ K L
this✝ : IsScalarTower Aₘ Bₘ L
this : IsFractionRing Bₘ L := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization (algebraMapSubmonoid B M) Bₘ L
\u22a2 (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)