Theorem: Ideal.spanIntNorm_map
Proof state before:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝¹⁰ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁹ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁸ : IsDomain Sₚ
this✝⁷ : IsDedekindDomain Sₚ
this✝⁶ : IsPrincipalIdealRing Rₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝² : Algebra Rₚ K := f.toAlgebra
this✝¹ : IsScalarTower R Rₚ K
this✝ : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this : Algebra Sₚ L := g.toAlgebra
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
haveI : IsScalarTower S Sₚ (FractionRing S) := IsScalarTower.of_algebraMap_eq'
  (by rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHom.comp_id])
Proof state after:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝¹¹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝¹⁰ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁹ : IsDomain Sₚ
this✝⁸ : IsDedekindDomain Sₚ
this✝⁷ : IsPrincipalIdealRing Rₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝³ : Algebra Rₚ K := f.toAlgebra
this✝² : IsScalarTower R Rₚ K
this✝¹ : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝ : Algebra Sₚ L := g.toAlgebra
this : IsScalarTower S Sₚ (FractionRing S)
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))