haveI := isIntegrallyClosed_of_isLocalization P.primeCompl P.primeCompl_le_nonZeroDivisors Rₚ
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✝³ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝² : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝¹ : IsDomain Sₚ
this✝ : IsDedekindDomain Sₚ
this : IsPrincipalIdealRing 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✝⁴ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝³ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝² : IsDomain Sₚ
this✝¹ : IsDedekindDomain Sₚ
this✝ : IsPrincipalIdealRing Sₚ
this : IsIntegrallyClosed Rₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)