Theorem: Algebra.intNorm_eq_zero
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✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
x : B
⊢ (intNorm A B) x = 0 ↔ x = 0
state_after: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (intNorm A B) x = 0 ↔ x = 0
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✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (intNorm A B) x = 0 ↔ x = 0
state_after: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (intNorm A B) x = 0 ↔ x = 0
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✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (intNorm A B) x = 0 ↔ x = 0
state_after: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
x : B
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ (intNorm A B) x = 0 ↔ x = 0
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✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
x : B
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ (intNorm A B) x = 0 ↔ x = 0
state_after: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing 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 (FractionRing A)) ((intNorm A B) x) = (algebraMap A (FractionRing A)) 0 ↔
    (algebraMap B (FractionRing B)) x = (algebraMap B (FractionRing B)) 0
tactic: rw [← (IsFractionRing.injective A (FractionRing A)).eq_iff,
    ← (IsFractionRing.injective B (FractionRing B)).eq_iff]
state_before: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³¹ : CommRing A
inst✝³⁰ : CommRing B
inst✝²⁹ : Algebra A B
inst✝²⁸ : Field K
inst✝²⁷ : Field L
inst✝²⁶ : Algebra A K
inst✝²⁵ : IsFractionRing A K
inst✝²⁴ : Algebra B L
inst✝²³ : Algebra K L
inst✝²² : Algebra A L
inst✝²¹ : IsScalarTower A B L
inst✝²⁰ : IsScalarTower A K L
inst✝¹⁹ : IsIntegralClosure B A L
inst✝¹⁸ : FiniteDimensional K L
Aₘ : Type ?u.632038
Bₘ : Type ?u.632041
inst✝¹⁷ : CommRing Aₘ
inst✝¹⁶ : CommRing Bₘ
inst✝¹⁵ : Algebra Aₘ Bₘ
inst✝¹⁴ : Algebra A Aₘ
inst✝¹³ : Algebra B Bₘ
inst✝¹² : Algebra A Bₘ
inst✝¹¹ : IsScalarTower A Aₘ Bₘ
inst✝¹⁰ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁹ : IsLocalization M Aₘ
inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing 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 (FractionRing A)) ((intNorm A B) x) = (algebraMap A (FractionRing A)) 0 ↔
    (algebraMap B (FractionRing B)) x = (algebraMap B (FractionRing B)) 0
state_after: no goals
tactic: simp only [algebraMap_intNorm_fractionRing, map_zero, norm_eq_zero_iff]