internlm/Lean-Github huggingface dataset:
Theorem: Ideal.spanIntNorm_map
Proof state before:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝² : IsIntegrallyClosed Rₚ
this✝¹ : NoZeroSMulDivisors Rₚ Sₚ
this✝ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this : Algebra Rₚ K := f.toAlgebra
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
have : IsScalarTower R Rₚ K := IsScalarTower.of_algebraMap_eq'
  (by rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHomCompTriple.comp_eq])
Proof state after:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝³ : IsIntegrallyClosed Rₚ
this✝² : NoZeroSMulDivisors Rₚ Sₚ
this✝¹ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝ : Algebra Rₚ K := f.toAlgebra
this : IsScalarTower R Rₚ K
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Theorem: Ideal.spanIntNorm_map
Proof state before:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝³ : IsIntegrallyClosed Rₚ
this✝² : NoZeroSMulDivisors Rₚ Sₚ
this✝¹ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝ : Algebra Rₚ K := f.toAlgebra
this : IsScalarTower R Rₚ K
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
letI := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
Proof state after:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝¹ : Algebra Rₚ K := f.toAlgebra
this✝ : IsScalarTower R Rₚ K
this : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Theorem: Ideal.spanIntNorm_map
Proof state before:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝¹ : Algebra Rₚ K := f.toAlgebra
this✝ : IsScalarTower R Rₚ K
this : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
let L := FractionRing S
Proof state after:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝¹ : Algebra Rₚ K := f.toAlgebra
this✝ : IsScalarTower R Rₚ K
this : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Theorem: Ideal.spanIntNorm_map
Proof state before:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝¹ : Algebra Rₚ K := f.toAlgebra
this✝ : IsScalarTower R Rₚ K
this : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
let g : Sₚ →+* L := IsLocalization.map _ (M := P') (T := S⁰) (RingHom.id S) h
Proof state after:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝¹ : Algebra Rₚ K := f.toAlgebra
this✝ : IsScalarTower R Rₚ K
this : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Theorem: Ideal.spanIntNorm_map
Proof state before:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝⁴ : IsIntegrallyClosed Rₚ
this✝³ : NoZeroSMulDivisors Rₚ Sₚ
this✝² : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝¹ : Algebra Rₚ K := f.toAlgebra
this✝ : IsScalarTower R Rₚ K
this : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
letI := g.toAlgebra
Proof state after:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝¹⁰ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁹ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁸ : IsDomain Sₚ
this✝⁷ : IsDedekindDomain Sₚ
this✝⁶ : IsPrincipalIdealRing Rₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝² : Algebra Rₚ K := f.toAlgebra
this✝¹ : IsScalarTower R Rₚ K
this✝ : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this : Algebra Sₚ L := g.toAlgebra
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Theorem: Ideal.spanIntNorm_map
Proof state before:
case e_s.e_a.a
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝¹⁸ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝¹⁷ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝¹⁶ : IsDomain Sₚ
this✝¹⁵ : IsDedekindDomain Sₚ
this✝¹⁴ : IsPrincipalIdealRing Rₚ
this✝¹³ : IsIntegrallyClosed Rₚ
this✝¹² : NoZeroSMulDivisors Rₚ Sₚ
this✝¹¹ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝¹⁰ : Algebra Rₚ K := f.toAlgebra
this✝⁹ : IsScalarTower R Rₚ K
this✝⁸ : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝⁷ : Algebra Sₚ L := g.toAlgebra
this✝⁶ : IsScalarTower S Sₚ (FractionRing S)
this✝⁵ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝⁴ : Algebra Rₚ L := ((algebraMap K L).comp f).toAlgebra
this✝³ : IsScalarTower Rₚ K L
this✝² : IsScalarTower Rₚ Sₚ L
this✝¹ : IsIntegralClosure Sₚ Rₚ L
this✝ : IsSeparable (FractionRing Rₚ) (FractionRing Sₚ)
this : Module.Free Rₚ Sₚ
⊢ (algebraMap Rₚ K)
      ((Algebra.intNorm (Localization.AtPrime P) (Localization P'))
        ((algebraMap Rₚ Sₚ) (IsPrincipal.generator (map (algebraMap R Rₚ) I)))) =
    (algebraMap Rₚ K)
      (IsPrincipal.generator (map (algebraMap R Rₚ) I) ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
rw [Algebra.algebraMap_intNorm (L := L), ← IsScalarTower.algebraMap_apply,
  IsScalarTower.algebraMap_apply Rₚ K L, Algebra.norm_algebraMap, map_pow]
Proof state after:
no goals
Theorem: Ideal.spanIntNorm_map
Proof state before:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝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 Rₚ
this✝² : IsIntegrallyClosed Rₚ
this✝¹ : NoZeroSMulDivisors Rₚ Sₚ
this✝ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this : Algebra Rₚ K := f.toAlgebra
⊢ algebraMap R K = (algebraMap Rₚ K).comp (algebraMap R Rₚ)
Tactic:
rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHomCompTriple.comp_eq]
Proof state after:
no goals
Theorem: Ideal.spanIntNorm_map
Proof state before:
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝¹⁰ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁹ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁸ : IsDomain Sₚ
this✝⁷ : IsDedekindDomain Sₚ
this✝⁶ : IsPrincipalIdealRing Rₚ
this✝⁵ : IsIntegrallyClosed Rₚ
this✝⁴ : NoZeroSMulDivisors Rₚ Sₚ
this✝³ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝² : Algebra Rₚ K := f.toAlgebra
this✝¹ : IsScalarTower R Rₚ K
this✝ : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this : Algebra Sₚ L := g.toAlgebra
⊢ algebraMap S (FractionRing S) = (algebraMap Sₚ (FractionRing S)).comp (algebraMap S Sₚ)
Tactic:
rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHom.comp_id]
Proof state after:
no goals
Theorem: Ideal.spanIntNorm_map
Proof state before:
case h.h
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝¹⁴ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝¹³ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝¹² : IsDomain Sₚ
this✝¹¹ : IsDedekindDomain Sₚ
this✝¹⁰ : IsPrincipalIdealRing Rₚ
this✝⁹ : IsIntegrallyClosed Rₚ
this✝⁸ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁷ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝⁶ : Algebra Rₚ K := f.toAlgebra
this✝⁵ : IsScalarTower R Rₚ K
this✝⁴ : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝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✝ : Algebra Rₚ L := ((algebraMap K L).comp f).toAlgebra
this : IsScalarTower Rₚ K L
⊢ (algebraMap Rₚ L).comp (algebraMap R Rₚ) = ((algebraMap Sₚ L).comp (algebraMap Rₚ Sₚ)).comp (algebraMap R Rₚ)
Tactic:
rw [RingHom.algebraMap_toAlgebra, RingHom.algebraMap_toAlgebra (R := Sₚ), RingHom.comp_assoc,
  RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq, IsScalarTower.algebraMap_eq R S Sₚ,
  IsLocalization.map_comp, RingHom.comp_id, ← RingHom.comp_assoc, IsLocalization.map_comp,
  RingHom.comp_id, ← IsScalarTower.algebraMap_eq, ← IsScalarTower.algebraMap_eq]
Proof state after:
no goals
Theorem: Ideal.spanIntNorm_map
Proof state before:
case he.h.a
R : Type u_1
inst✝¹⁶ : CommRing R
S : Type u_2
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
inst✝¹³ : IsIntegrallyClosed R
inst✝¹² : IsDomain R
inst✝¹¹ : IsDomain S
inst✝¹⁰ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : IsSeparable (FractionRing R) (FractionRing S)
inst✝⁷ : IsDomain R
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Finite R S
inst✝² : Module.Free R S
I : Ideal R
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_2 := Localization P'
this✝¹⁶ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝¹⁵ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝¹⁴ : IsDomain Sₚ
this✝¹³ : IsDedekindDomain Sₚ
this✝¹² : IsPrincipalIdealRing Rₚ
this✝¹¹ : IsIntegrallyClosed Rₚ
this✝¹⁰ : NoZeroSMulDivisors Rₚ Sₚ
this✝⁹ : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
this✝⁸ : Algebra Rₚ K := f.toAlgebra
this✝⁷ : IsScalarTower R Rₚ K
this✝⁶ : IsFractionRing Rₚ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P.primeCompl Rₚ K
L : Type u_2 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
this✝⁵ : Algebra Sₚ L := g.toAlgebra
this✝⁴ : IsScalarTower S Sₚ (FractionRing S)
this✝³ : IsFractionRing Sₚ (FractionRing S) :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization P' Sₚ (FractionRing S)
this✝² : Algebra Rₚ L := ((algebraMap K L).comp f).toAlgebra
this✝¹ : IsScalarTower Rₚ K L
this✝ : IsScalarTower Rₚ Sₚ L
this : IsIntegralClosure Sₚ Rₚ L
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]
Proof state after:
no goals