-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
@[simp]
lemma Algebra.algebraMap_intNorm_fractionRing (x : B) :
    algebraMap A (FractionRing A) (Algebra.intNorm A B x) =
      Algebra.norm (FractionRing A) (algebraMap B (FractionRing B) x)

Proof state:
A : Type u_1
B : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDomain 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)
⊢ (algebraMap A (FractionRing A)) ((intNorm A B) x) = (norm (FractionRing A)) ((algebraMap B (FractionRing B)) x)

Tactic:
haveI : FiniteDimensional (FractionRing A) (FractionRing B) :=
  Module.Finite_of_isLocalization A B _ _ A⁰

Result:
A : Type u_1
B : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDomain 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) = (norm (FractionRing A)) ((algebraMap B (FractionRing B)) x)

Proof state:
A : Type u_1
B : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDomain 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
⊢ (algebraMap A (FractionRing A)) ((intNorm A B) x) = (norm (FractionRing A)) ((algebraMap B (FractionRing B)) x)

Tactic:
haveI : IsIntegralClosure B A (FractionRing B) :=
  IsIntegralClosure.of_isIntegrallyClosed _ _ _

Result:
A : Type u_1
B : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDomain 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)
⊢ (algebraMap A (FractionRing A)) ((intNorm A B) x) = (norm (FractionRing A)) ((algebraMap B (FractionRing B)) x)

Proof state:
A : Type u_1
B : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDomain 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) = (norm (FractionRing A)) ((algebraMap B (FractionRing B)) x)

Tactic:
exact Algebra.map_intNormAux x

Result:
no goals

Proof state:
A : Type u_1
B : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDomain 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)
⊢ (algebraMap A (FractionRing A)) ((intNorm A B) x) = (norm (FractionRing A)) ((algebraMap B (FractionRing B)) x)

Tactic:
haveI : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B) :=
  IsIntegralClosure.isLocalization _ (FractionRing A) _ _

Result:
A : Type u_1
B : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDomain 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)
⊢ (algebraMap A (FractionRing A)) ((intNorm A B) x) = (norm (FractionRing A)) ((algebraMap B (FractionRing B)) x)