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✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : IsScalarTower R Rₚ Sₚ
⊢ 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 h : P' ≤ S⁰ :=
  map_le_nonZeroDivisors_of_injective _ (NoZeroSMulDivisors.algebraMap_injective _ _)
    P.primeCompl_le_nonZeroDivisors
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✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
⊢ 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✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
haveI : IsDomain Sₚ := IsLocalization.isDomain_localization 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✝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 (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✝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 (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
haveI : IsDedekindDomain Sₚ := IsLocalization.isDedekindDomain 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✝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 (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✝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 (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
haveI : IsPrincipalIdealRing Rₚ := IsLocalization.AtPrime.PID_of_dedekind_domain P Rₚ
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✝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 Rₚ
⊢ 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✝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 Rₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
haveI := isIntegrallyClosed_of_isLocalization P.primeCompl P.primeCompl_le_nonZeroDivisors Rₚ
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✝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 Rₚ
this : IsIntegrallyClosed Rₚ
⊢ 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✝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 Rₚ
this : IsIntegrallyClosed Rₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
haveI := NoZeroSMulDivisors_of_isLocalization R S Rₚ Sₚ _ P.primeCompl_le_nonZeroDivisors
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✝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 Rₚ
this✝ : IsIntegrallyClosed Rₚ
this : NoZeroSMulDivisors Rₚ Sₚ
⊢ 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✝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 Rₚ
this✝ : IsIntegrallyClosed Rₚ
this : NoZeroSMulDivisors Rₚ Sₚ
⊢ map (algebraMap R (Localization.AtPrime P)) (spanIntNorm R (map (algebraMap R S) I)) =
    map (algebraMap R (Localization.AtPrime P)) (I ^ FiniteDimensional.finrank (FractionRing R) (FractionRing S))
Tactic:
haveI := Module.Finite_of_isLocalization R S Rₚ Sₚ P.primeCompl
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✝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 Rₚ
this✝¹ : IsIntegrallyClosed Rₚ
this✝ : NoZeroSMulDivisors Rₚ Sₚ
this : Module.Finite Rₚ Sₚ
⊢ 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✝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 Rₚ
this✝¹ : IsIntegrallyClosed Rₚ
this✝ : NoZeroSMulDivisors Rₚ Sₚ
this : Module.Finite Rₚ Sₚ
⊢ 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 K := FractionRing R
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✝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 Rₚ
this✝¹ : IsIntegrallyClosed Rₚ
this✝ : NoZeroSMulDivisors Rₚ Sₚ
this : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
⊢ 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✝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 Rₚ
this✝¹ : IsIntegrallyClosed Rₚ
this✝ : NoZeroSMulDivisors Rₚ Sₚ
this : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
⊢ 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 f : Rₚ →+* K := IsLocalization.map _ (T := R⁰) (RingHom.id R) P.primeCompl_le_nonZeroDivisors
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✝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 Rₚ
this✝¹ : IsIntegrallyClosed Rₚ
this✝ : NoZeroSMulDivisors Rₚ Sₚ
this : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
⊢ 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✝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 Rₚ
this✝¹ : IsIntegrallyClosed Rₚ
this✝ : NoZeroSMulDivisors Rₚ Sₚ
this : Module.Finite Rₚ Sₚ
K : Type u_1 := FractionRing R
f : Rₚ →+* K := IsLocalization.map K (RingHom.id R) ⋯
⊢ 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 := f.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✝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))