-- Mathlib/RingTheory/DedekindDomain/PID.lean
theorem IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime [IsDomain S]
    {P : Ideal Sₚ} (hP : IsPrime P) (hP0 : P ≠ ⊥) :
    P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)

Proof state:
case intro.intro.intro.intro
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : NoZeroSMulDivisors R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)

Tactic:
have : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥ := by
  rw [Submodule.ne_bot_iff] at hp0 ⊢
  obtain ⟨x, x_mem, x_ne⟩ := hp0
  exact
    ⟨algebraMap _ _ x, (IsLocalization.AtPrime.to_map_mem_maximal_iff _ _ _).mpr x_mem,
      IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors _ p.primeCompl_le_nonZeroDivisors
        (mem_nonZeroDivisors_of_ne_zero x_ne)⟩

Result:
case intro.intro.intro.intro
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : NoZeroSMulDivisors R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)

Proof state:
case intro.intro.intro.intro
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : NoZeroSMulDivisors R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Function.Injective ⇑((algebraMap S Sₚ).comp (algebraMap R S))

Tactic:
exact
  (IsLocalization.injective Sₚ non_zero_div).comp (FaithfulSMul.algebraMap_injective _ _)

Result:
no goals

Proof state:
case intro.intro.intro.intro
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : NoZeroSMulDivisors R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Function.Injective ⇑(algebraMap R Sₚ)

Tactic:
rw [IsScalarTower.algebraMap_eq R S Sₚ]

Result:
case intro.intro.intro.intro
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : NoZeroSMulDivisors R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Function.Injective ⇑((algebraMap S Sₚ).comp (algebraMap R S))

Proof state:
case intro.intro
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : NoZeroSMulDivisors R S
inst✝⁸ : Module.Finite R S
p : Ideal R
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
x : R
x_mem : x ∈ p
x_ne : x ≠ 0
⊢ ∃ x ∈ IsLocalRing.maximalIdeal (Localization.AtPrime p), x ≠ 0

Tactic:
exact
  ⟨algebraMap _ _ x, (IsLocalization.AtPrime.to_map_mem_maximal_iff _ _ _).mpr x_mem,
    IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors _ p.primeCompl_le_nonZeroDivisors
      (mem_nonZeroDivisors_of_ne_zero x_ne)⟩

Result:
no goals