haveI := Module.Finite_of_isLocalization R S Rₚ Sₚ P.primeCompl
case neg
R : Type u_1
inst✝¹⁵ : CommRing R
S : Type u_2
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
inst✝¹² : IsIntegrallyClosed R
inst✝¹¹ : IsDomain R
inst✝¹⁰ : IsDomain S
inst✝⁹ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁶ : IsDomain R
inst✝⁵ : IsDomain S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
inst✝² : Module.Finite R S
inst✝¹ : Module.Free R S
I J : Ideal S
inst✝ : Nontrivial R
h✝ : Nontrivial S
P : Ideal R
hP : P.IsMaximal
hP0 : ¬P = ⊥
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝⁵ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁴ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝³ : IsDomain Sₚ
this✝² : IsDedekindDomain Sₚ
this✝¹ : IsPrincipalIdealRing Sₚ
this✝ : IsIntegrallyClosed Rₚ
this : NoZeroSMulDivisors Rₚ Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
case neg
R : Type u_1
inst✝¹⁵ : CommRing R
S : Type u_2
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
inst✝¹² : IsIntegrallyClosed R
inst✝¹¹ : IsDomain R
inst✝¹⁰ : IsDomain S
inst✝⁹ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁶ : IsDomain R
inst✝⁵ : IsDomain S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
inst✝² : Module.Finite R S
inst✝¹ : Module.Free R S
I J : Ideal S
inst✝ : Nontrivial R
h✝ : Nontrivial S
P : Ideal R
hP : P.IsMaximal
hP0 : ¬P = ⊥
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝⁶ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁵ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁴ : IsDomain Sₚ
this✝³ : IsDedekindDomain Sₚ
this✝² : IsPrincipalIdealRing Sₚ
this✝¹ : IsIntegrallyClosed Rₚ
this✝ : NoZeroSMulDivisors Rₚ Sₚ
this : Module.Finite Rₚ Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)