/-- If `p` is a prime in the Dedekind domain `R`, `S` an extension of `R` and `Sₚ` the localization
of `S` at `p`, then all primes in `Sₚ` are factors of the image of `p` in `Sₚ`. -/
theorem IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime [DecidableEq (Ideal Sₚ)]
    {P : Ideal Sₚ} (hP : IsPrime P) (hP0 : P ≠ ⊥) :
    P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p) := by
  have non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰ :=
    map_le_nonZeroDivisors_of_injective _ (NoZeroSMulDivisors.algebraMap_injective _ _)
      p.primeCompl_le_nonZeroDivisors
  letI : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
  haveI : IsScalarTower R (Localization.AtPrime p) Sₚ :=
    IsScalarTower.of_algebraMap_eq fun x => by
      -- Porting note: replaced `erw` with a `rw` followed by `exact` to help infer implicits
      rw [IsScalarTower.algebraMap_apply R S]
      exact (IsLocalization.map_eq (T := Algebra.algebraMapSubmonoid S (primeCompl p))
        (Submonoid.le_comap_map _) x).symm
  obtain ⟨pid, p', ⟨hp'0, hp'p⟩, hpu⟩ :=
    (DiscreteValuationRing.iff_pid_with_one_nonzero_prime (Localization.AtPrime p)).mp
      (IsLocalization.AtPrime.discreteValuationRing_of_dedekind_domain R hp0 _)
  have : LocalRing.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)⟩
  rw [← Multiset.singleton_le, ← normalize_eq P, ←
    normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, ←
    dvd_iff_normalizedFactors_le_normalizedFactors hP0, dvd_iff_le,
    IsScalarTower.algebraMap_eq R (Localization.AtPrime p) Sₚ, ← Ideal.map_map,
    Localization.AtPrime.map_eq_maximalIdeal, Ideal.map_le_iff_le_comap,
    hpu (LocalRing.maximalIdeal _) ⟨this, _⟩, hpu (comap _ _) ⟨_, _⟩]
  · have hRS : Algebra.IsIntegral R S := isIntegral_of_noetherian inferInstance
    exact mt (Ideal.eq_bot_of_comap_eq_bot (isIntegral_localization hRS)) hP0
  · exact Ideal.comap_isPrime (algebraMap (Localization.AtPrime p) Sₚ) P
  · exact (LocalRing.maximalIdeal.isMaximal _).isPrime
  · rw [Ne.def, zero_eq_bot, Ideal.map_eq_bot_iff_of_injective]
    · assumption
    rw [IsScalarTower.algebraMap_eq R S Sₚ]
    

Proof state:
case intro.intro.intro.intro
R : Type u_1
inst✝¹⁶ : CommRing R
inst✝¹⁵ : IsDomain R
inst✝¹⁴ : IsDedekindDomain R
S : Type u_2
inst✝¹³ : CommRing S
inst✝¹² : IsDomain S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Module.Free R S
inst✝⁹ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁸ : IsPrime p
Sₚ : Type u_3
inst✝⁷ : CommRing Sₚ
inst✝⁶ : Algebra S Sₚ
inst✝⁵ : IsLocalization (Algebra.algebraMapSubmonoid S (primeCompl p)) Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : IsScalarTower R S Sₚ
inst✝² : IsDomain Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : DecidableEq (Ideal Sₚ)
P : Ideal Sₚ
hP : IsPrime P
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S (primeCompl p) ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra (primeCompl p) 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 ≠ ⊥ ∧ IsPrime P) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : IsPrime p'
this : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Function.Injective ⇑(RingHom.comp (algebraMap S Sₚ) (algebraMap R S))

Next tactic:
exact
      (IsLocalization.injective Sₚ non_zero_div).comp (NoZeroSMulDivisors.algebraMap_injective _ _)