Theorem: Ideal.spanIntNorm_mul (FltRegular/NumberTheory/IdealNorm.lean)

State before:
R : Type u_1
inst✝¹⁴ : CommRing R
S : Type u_2
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
inst✝¹¹ : IsIntegrallyClosed R
inst✝¹⁰ : IsDomain R
inst✝⁹ : IsDomain S
inst✝⁸ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁷ : IsIntegrallyClosed S
inst✝⁶ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁵ : IsDomain R
inst✝⁴ : IsDomain S
inst✝³ : IsDedekindDomain R
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Finite R S
inst✝ : Module.Free R S
I J : Ideal S
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J
Tactic:
nontriviality R
State after:
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
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J

State before:
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
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J
Tactic:
cases subsingleton_or_nontrivial S
State after:
case inl
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✝ : Subsingleton S
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J

case inr
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
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J

State before:
case inr
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
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J
Tactic:
refine eq_of_localization_maximal ?_
State after:
case inr
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) (x : P.IsMaximal),
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
      map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
case inr
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) (x : P.IsMaximal),
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
      map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
intro P hP
State after:
case inr
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
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
case inr
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
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
by_cases hP0 : P = ⊥
State after:
case pos
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 = ⊥
⊢ 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 = ⊥
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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 = ⊥
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
let P' := Algebra.algebraMapSubmonoid S P.primeCompl
State after:
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
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
let Rₚ := Localization.AtPrime P
State after:
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
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
let Sₚ := Localization P'
State after:
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'
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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'
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
letI : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
State after:
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 : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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 : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
haveI : IsScalarTower R Rₚ Sₚ :=
  IsScalarTower.of_algebraMap_eq (fun x =>
    (IsLocalization.map_eq (T := P') (Q := Localization P') P.primeCompl.le_comap_map x).symm)
State after:
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✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : IsScalarTower R Rₚ Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : IsScalarTower R Rₚ Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
have h : P' ≤ S⁰ :=
  map_le_nonZeroDivisors_of_injective _ (NoZeroSMulDivisors.algebraMap_injective _ _)
    P.primeCompl_le_nonZeroDivisors
State after:
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✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
haveI : IsDomain Sₚ := IsLocalization.isDomain_localization h
State after:
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✝¹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this : IsDomain Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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✝¹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this : IsDomain Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
haveI : IsDedekindDomain Sₚ := IsLocalization.isDedekindDomain S h _
State after:
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✝² : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝¹ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝ : IsDomain Sₚ
this : IsDedekindDomain Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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✝² : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝¹ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝ : IsDomain Sₚ
this : IsDedekindDomain Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
haveI : IsPrincipalIdealRing Sₚ :=
  IsDedekindDomain.isPrincipalIdealRing_localization_over_prime S P hP0
State after:
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)

State before:
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)
Tactic:
haveI := isIntegrallyClosed_of_isLocalization P.primeCompl P.primeCompl_le_nonZeroDivisors Rₚ
State after:
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)

State before:
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)
Tactic:
haveI := NoZeroSMulDivisors_of_isLocalization R S Rₚ Sₚ _ P.primeCompl_le_nonZeroDivisors
State after:
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)

State before:
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)
Tactic:
haveI := Module.Finite_of_isLocalization R S Rₚ Sₚ P.primeCompl
State after:
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)

State before:
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)
Tactic:
let L := FractionRing S
State after:
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
L : Type u_2 := FractionRing S
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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
L : Type u_2 := FractionRing S
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
let g : Sₚ →+* L := IsLocalization.map _ (M := P') (T := S⁰) (RingHom.id S) h
State after:
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
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
letI := g.toAlgebra
State after:
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✝⁷ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁶ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁵ : IsDomain Sₚ
this✝⁴ : IsDedekindDomain Sₚ
this✝³ : IsPrincipalIdealRing Sₚ
this✝² : IsIntegrallyClosed Rₚ
this✝¹ : NoZeroSMulDivisors Rₚ Sₚ
this✝ : Module.Finite Rₚ Sₚ
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 (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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✝⁷ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁶ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁵ : IsDomain Sₚ
this✝⁴ : IsDedekindDomain Sₚ
this✝³ : IsPrincipalIdealRing Sₚ
this✝² : IsIntegrallyClosed Rₚ
this✝¹ : NoZeroSMulDivisors Rₚ Sₚ
this✝ : Module.Finite Rₚ Sₚ
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 (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
haveI : IsScalarTower S Sₚ (FractionRing S) := IsScalarTower.of_algebraMap_eq'
  (by rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHom.comp_id])
State after:
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✝⁸ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁷ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁶ : IsDomain Sₚ
this✝⁵ : IsDedekindDomain Sₚ
this✝⁴ : IsPrincipalIdealRing Sₚ
this✝³ : IsIntegrallyClosed Rₚ
this✝² : NoZeroSMulDivisors Rₚ Sₚ
this✝¹ : Module.Finite Rₚ Sₚ
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 (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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✝⁸ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁷ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁶ : IsDomain Sₚ
this✝⁵ : IsDedekindDomain Sₚ
this✝⁴ : IsPrincipalIdealRing Sₚ
this✝³ : IsIntegrallyClosed Rₚ
this✝² : NoZeroSMulDivisors Rₚ Sₚ
this✝¹ : Module.Finite Rₚ Sₚ
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 (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
letI := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization
  P' Sₚ (FractionRing S)
State after:
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✝⁹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁸ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁷ : IsDomain Sₚ
this✝⁶ : IsDedekindDomain Sₚ
this✝⁵ : IsPrincipalIdealRing Sₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝¹ : Algebra Sₚ L := g.toAlgebra
this✝ : IsScalarTower S Sₚ (FractionRing S)
this : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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✝⁹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁸ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁷ : IsDomain Sₚ
this✝⁶ : IsDedekindDomain Sₚ
this✝⁵ : IsPrincipalIdealRing Sₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝¹ : Algebra Sₚ L := g.toAlgebra
this✝ : IsScalarTower S Sₚ (FractionRing S)
this : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
haveI : IsSeparable (FractionRing Rₚ) (FractionRing Sₚ) := by
  haveI : IsScalarTower R Rₚ (FractionRing R) := IsScalarTower.of_algebraMap_eq'
    (by rw [RingHom.algebraMap_toAlgebra, IsLocalization.lift_comp])
  letI := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization
    P.primeCompl Rₚ (FractionRing R)
  apply IsSeparable.of_equiv_equiv (FractionRing.algEquiv Rₚ (FractionRing R)).symm.toRingEquiv
    (FractionRing.algEquiv Sₚ (FractionRing S)).symm.toRingEquiv
  apply IsLocalization.ringHom_ext R⁰
  ext
  simp only [AlgEquiv.toRingEquiv_eq_coe, RingHom.coe_comp,
    RingHom.coe_coe, Function.comp_apply, ← IsScalarTower.algebraMap_apply]
  simp only [IsScalarTower.algebraMap_apply R Rₚ (FractionRing R), AlgEquiv.coe_ringEquiv,
    AlgEquiv.commutes, IsScalarTower.algebraMap_apply R S L,
    IsScalarTower.algebraMap_apply S Sₚ L]
  simp only [← IsScalarTower.algebraMap_apply]
  rw [IsScalarTower.algebraMap_apply R Rₚ (FractionRing Rₚ), ← IsScalarTower.algebraMap_apply Rₚ,
    ← IsScalarTower.algebraMap_apply]
State after:
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✝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 Sₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝² : Algebra Sₚ L := g.toAlgebra
this✝¹ : IsScalarTower S Sₚ (FractionRing S)
this✝ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this : IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)

State before:
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✝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 Sₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝² : Algebra Sₚ L := g.toAlgebra
this✝¹ : IsScalarTower S Sₚ (FractionRing S)
this✝ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this : IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
simp only [Ideal.map_mul, ← spanIntNorm_localization (R := R) (S := S)
  (Rₘ := Localization.AtPrime P) (Sₘ := Localization P') _ _ P.primeCompl_le_nonZeroDivisors]
State after:
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✝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 Sₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝² : Algebra Sₚ L := g.toAlgebra
this✝¹ : IsScalarTower S Sₚ (FractionRing S)
this✝ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this : IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)
⊢ spanIntNorm (Localization.AtPrime P)
      (map (algebraMap S (Localization P')) I * map (algebraMap S (Localization P')) J) =
    spanIntNorm (Localization.AtPrime P) (map (algebraMap S (Localization P')) I) *
      spanIntNorm (Localization.AtPrime P) (map (algebraMap S (Localization P')) J)

State before:
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✝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 Sₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝² : Algebra Sₚ L := g.toAlgebra
this✝¹ : IsScalarTower S Sₚ (FractionRing S)
this✝ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this : IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)
⊢ spanIntNorm (Localization.AtPrime P)
      (map (algebraMap S (Localization P')) I * map (algebraMap S (Localization P')) J) =
    spanIntNorm (Localization.AtPrime P) (map (algebraMap S (Localization P')) I) *
      spanIntNorm (Localization.AtPrime P) (map (algebraMap S (Localization P')) J)
Tactic:
rw [← (I.map _).span_singleton_generator, ← (J.map _).span_singleton_generator,
  span_singleton_mul_span_singleton, spanIntNorm_singleton, spanIntNorm_singleton,
    spanIntNorm_singleton, span_singleton_mul_span_singleton, _root_.map_mul]
State after:
no goals

State before:
case inl
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✝ : Subsingleton S
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J
Tactic:
have : ∀ I : Ideal S, I = ⊤ := fun I => Subsingleton.elim I ⊤
State after:
case inl
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✝ : Subsingleton S
this : ∀ (I : Ideal S), I = ⊤
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J

State before:
case inl
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✝ : Subsingleton S
this : ∀ (I : Ideal S), I = ⊤
⊢ spanIntNorm R (I * J) = spanIntNorm R I * spanIntNorm R J
Tactic:
simp [this I, this J, this (I * J)]
State after:
no goals

State before:
case pos
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 = ⊥
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
subst hP0
State after:
case pos
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
hP : ⊥.IsMaximal
⊢ map (algebraMap R (Localization.AtPrime ⊥)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime ⊥)) (spanIntNorm R I * spanIntNorm R J)

State before:
case pos
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
hP : ⊥.IsMaximal
⊢ map (algebraMap R (Localization.AtPrime ⊥)) (spanIntNorm R (I * J)) =
    map (algebraMap R (Localization.AtPrime ⊥)) (spanIntNorm R I * spanIntNorm R J)
Tactic:
rw [spanIntNorm_mul_of_bot_or_top]
State after:
case pos.eq_bot_or_top
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
hP : ⊥.IsMaximal
⊢ ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤

State before:
case pos.eq_bot_or_top
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
hP : ⊥.IsMaximal
⊢ ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
Tactic:
intro I
State after:
case pos.eq_bot_or_top
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
hP : ⊥.IsMaximal
I : Ideal R
⊢ I = ⊥ ∨ I = ⊤

State before:
case pos.eq_bot_or_top
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
hP : ⊥.IsMaximal
I : Ideal R
⊢ I = ⊥ ∨ I = ⊤
Tactic:
refine or_iff_not_imp_right.mpr fun hI => ?_
State after:
case pos.eq_bot_or_top
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
hP : ⊥.IsMaximal
I : Ideal R
hI : ¬I = ⊤
⊢ I = ⊥

State before:
case pos.eq_bot_or_top
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
hP : ⊥.IsMaximal
I : Ideal R
hI : ¬I = ⊤
⊢ I = ⊥
Tactic:
exact (hP.eq_of_le hI bot_le).symm
State after:
no goals

State before:
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✝⁷ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁶ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁵ : IsDomain Sₚ
this✝⁴ : IsDedekindDomain Sₚ
this✝³ : IsPrincipalIdealRing Sₚ
this✝² : IsIntegrallyClosed Rₚ
this✝¹ : NoZeroSMulDivisors Rₚ Sₚ
this✝ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this : Algebra Sₚ L := g.toAlgebra
⊢ algebraMap S (FractionRing S) = (algebraMap Sₚ (FractionRing S)).comp (algebraMap S Sₚ)
Tactic:
rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHom.comp_id]
State after:
no goals

State before:
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✝⁹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁸ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁷ : IsDomain Sₚ
this✝⁶ : IsDedekindDomain Sₚ
this✝⁵ : IsPrincipalIdealRing Sₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝¹ : Algebra Sₚ L := g.toAlgebra
this✝ : IsScalarTower S Sₚ (FractionRing S)
this : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
⊢ IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)
Tactic:
haveI : IsScalarTower R Rₚ (FractionRing R) := IsScalarTower.of_algebraMap_eq'
  (by rw [RingHom.algebraMap_toAlgebra, IsLocalization.lift_comp])
State after:
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✝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 Sₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝² : Algebra Sₚ L := g.toAlgebra
this✝¹ : IsScalarTower S Sₚ (FractionRing S)
this✝ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this : IsScalarTower R Rₚ (FractionRing R)
⊢ IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)

State before:
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✝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 Sₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝² : Algebra Sₚ L := g.toAlgebra
this✝¹ : IsScalarTower S Sₚ (FractionRing S)
this✝ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this : IsScalarTower R Rₚ (FractionRing R)
⊢ IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)
Tactic:
letI := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization
  P.primeCompl Rₚ (FractionRing R)
State after:
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
⊢ IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)

State before:
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
⊢ IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)
Tactic:
apply IsSeparable.of_equiv_equiv (FractionRing.algEquiv Rₚ (FractionRing R)).symm.toRingEquiv
  (FractionRing.algEquiv Sₚ (FractionRing S)).symm.toRingEquiv
State after:
case he
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ)).comp ↑(FractionRing.algEquiv Rₚ (FractionRing R)).symm.toRingEquiv =
    (↑(FractionRing.algEquiv Sₚ (FractionRing S)).symm.toRingEquiv).comp (algebraMap (FractionRing R) (FractionRing S))

State before:
case he
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ)).comp ↑(FractionRing.algEquiv Rₚ (FractionRing R)).symm.toRingEquiv =
    (↑(FractionRing.algEquiv Sₚ (FractionRing S)).symm.toRingEquiv).comp (algebraMap (FractionRing R) (FractionRing S))
Tactic:
apply IsLocalization.ringHom_ext R⁰
State after:
case he.h
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
⊢ ((algebraMap (FractionRing Rₚ) (FractionRing Sₚ)).comp
          ↑(FractionRing.algEquiv Rₚ (FractionRing R)).symm.toRingEquiv).comp
      (algebraMap R (FractionRing R)) =
    ((↑(FractionRing.algEquiv Sₚ (FractionRing S)).symm.toRingEquiv).comp
          (algebraMap (FractionRing R) (FractionRing S))).comp
      (algebraMap R (FractionRing R))

State before:
case he.h
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
⊢ ((algebraMap (FractionRing Rₚ) (FractionRing Sₚ)).comp
          ↑(FractionRing.algEquiv Rₚ (FractionRing R)).symm.toRingEquiv).comp
      (algebraMap R (FractionRing R)) =
    ((↑(FractionRing.algEquiv Sₚ (FractionRing S)).symm.toRingEquiv).comp
          (algebraMap (FractionRing R) (FractionRing S))).comp
      (algebraMap R (FractionRing R))
Tactic:
ext
State after:
case he.h.a
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
x✝ : R
⊢ (((algebraMap (FractionRing Rₚ) (FractionRing Sₚ)).comp
            ↑(FractionRing.algEquiv Rₚ (FractionRing R)).symm.toRingEquiv).comp
        (algebraMap R (FractionRing R)))
      x✝ =
    (((↑(FractionRing.algEquiv Sₚ (FractionRing S)).symm.toRingEquiv).comp
            (algebraMap (FractionRing R) (FractionRing S))).comp
        (algebraMap R (FractionRing R)))
      x✝

State before:
case he.h.a
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
x✝ : R
⊢ (((algebraMap (FractionRing Rₚ) (FractionRing Sₚ)).comp
            ↑(FractionRing.algEquiv Rₚ (FractionRing R)).symm.toRingEquiv).comp
        (algebraMap R (FractionRing R)))
      x✝ =
    (((↑(FractionRing.algEquiv Sₚ (FractionRing S)).symm.toRingEquiv).comp
            (algebraMap (FractionRing R) (FractionRing S))).comp
        (algebraMap R (FractionRing R)))
      x✝
Tactic:
simp only [AlgEquiv.toRingEquiv_eq_coe, RingHom.coe_comp,
  RingHom.coe_coe, Function.comp_apply, ← IsScalarTower.algebraMap_apply]
State after:
case he.h.a
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
x✝ : R
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ))
      (↑(FractionRing.algEquiv Rₚ (FractionRing R)).symm ((algebraMap R (FractionRing R)) x✝)) =
    ↑(FractionRing.algEquiv Sₚ (FractionRing S)).symm ((algebraMap R (FractionRing S)) x✝)

State before:
case he.h.a
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
x✝ : R
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ))
      (↑(FractionRing.algEquiv Rₚ (FractionRing R)).symm ((algebraMap R (FractionRing R)) x✝)) =
    ↑(FractionRing.algEquiv Sₚ (FractionRing S)).symm ((algebraMap R (FractionRing S)) x✝)
Tactic:
simp only [IsScalarTower.algebraMap_apply R Rₚ (FractionRing R), AlgEquiv.coe_ringEquiv,
  AlgEquiv.commutes, IsScalarTower.algebraMap_apply R S L,
  IsScalarTower.algebraMap_apply S Sₚ L]
State after:
case he.h.a
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
x✝ : R
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ)) ((algebraMap Rₚ (FractionRing Rₚ)) ((algebraMap R Rₚ) x✝)) =
    (algebraMap Sₚ (FractionRing Sₚ)) ((algebraMap S Sₚ) ((algebraMap R S) x✝))

State before:
case he.h.a
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
x✝ : R
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ)) ((algebraMap Rₚ (FractionRing Rₚ)) ((algebraMap R Rₚ) x✝)) =
    (algebraMap Sₚ (FractionRing Sₚ)) ((algebraMap S Sₚ) ((algebraMap R S) x✝))
Tactic:
simp only [← IsScalarTower.algebraMap_apply]
State after:
case he.h.a
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
x✝ : R
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ)) ((algebraMap R (FractionRing Rₚ)) x✝) =
    (algebraMap R (FractionRing Sₚ)) x✝

State before:
case he.h.a
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✝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 Sₚ
this✝⁶ : IsIntegrallyClosed Rₚ
this✝⁵ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁴ : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝³ : Algebra Sₚ L := g.toAlgebra
this✝² : IsScalarTower S Sₚ (FractionRing S)
this✝¹ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝ : IsScalarTower R Rₚ (FractionRing R)
this : IsFractionRing Rₚ (FractionRing R) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ (FractionRing R)
x✝ : R
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ)) ((algebraMap R (FractionRing Rₚ)) x✝) =
    (algebraMap R (FractionRing Sₚ)) x✝
Tactic:
rw [IsScalarTower.algebraMap_apply R Rₚ (FractionRing Rₚ), ← IsScalarTower.algebraMap_apply Rₚ,
  ← IsScalarTower.algebraMap_apply]
State after:
no goals

State before:
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✝⁹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁸ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁷ : IsDomain Sₚ
this✝⁶ : IsDedekindDomain Sₚ
this✝⁵ : IsPrincipalIdealRing Sₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝¹ : Algebra Sₚ L := g.toAlgebra
this✝ : IsScalarTower S Sₚ (FractionRing S)
this : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
⊢ algebraMap R (FractionRing R) = (algebraMap Rₚ (FractionRing R)).comp (algebraMap R Rₚ)
Tactic:
rw [RingHom.algebraMap_toAlgebra, IsLocalization.lift_comp]
State after:
no goals