haveI : NoZeroSMulDivisors Rₚ Sₚ := by
  rw [NoZeroSMulDivisors.iff_algebraMap_injective, RingHom.injective_iff_ker_eq_bot,
    RingHom.ker_eq_bot_iff_eq_zero]
  intro x hx
  obtain ⟨x, s, rfl⟩ := IsLocalization.mk'_surjective p.primeCompl x
  simp only [RingHom.algebraMap_toAlgebra, IsLocalization.map_mk', IsLocalization.mk'_eq_zero_iff,
    mul_eq_zero, Subtype.exists, exists_prop] at hx ⊢
  obtain ⟨_, ⟨a, ha, rfl⟩, H⟩ := hx
  simp only [(injective_iff_map_eq_zero' _).mp (NoZeroSMulDivisors.algebraMap_injective R S)] at H
  refine ⟨a, ha, H⟩
R : Type u_1
S : Type u_2
inst✝²⁰ : CommRing R
inst✝¹⁹ : CommRing S
inst✝¹⁸ : Algebra R S
p : Ideal R
inst✝¹⁷ : p.IsMaximal
Rₚ✝ : Type ?u.335608
Sₚ✝ : Type ?u.335611
inst✝¹⁶ : CommRing Rₚ✝
inst✝¹⁵ : CommRing Sₚ✝
inst✝¹⁴ : Algebra R Rₚ✝
inst✝¹³ : IsLocalization.AtPrime Rₚ✝ p
inst✝¹² : LocalRing Rₚ✝
inst✝¹¹ : CommRing Sₚ✝
inst✝¹⁰ : Algebra S Sₚ✝
inst✝⁹ : Algebra R Sₚ✝
inst✝⁸ : Algebra Rₚ✝ Sₚ✝
inst✝⁷ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ✝
inst✝⁶ : IsScalarTower R S Sₚ✝
inst✝⁵ : IsScalarTower R Rₚ✝ Sₚ✝
x : S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDomain S
inst✝² : NoZeroSMulDivisors R S
inst✝¹ : Module.Finite R S
inst✝ : IsIntegrallyClosed S
Rₚ : Type u_1 := Localization.AtPrime p
Sₚ : Type u_2 := Localization (algebraMapSubmonoid S p.primeCompl)
this✝² : Algebra Rₚ Sₚ := localizationAlgebra p.primeCompl S
this✝¹ : IsScalarTower R Rₚ Sₚ
this✝ : IsLocalization (Submonoid.map (algebraMap R S) p.primeCompl) Sₚ
e : algebraMapSubmonoid S p.primeCompl ≤ S⁰
this : IsDomain Sₚ
⊢ (trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) = (Ideal.Quotient.mk p) ((intTrace R S) x)
R : Type u_1
S : Type u_2
inst✝²⁰ : CommRing R
inst✝¹⁹ : CommRing S
inst✝¹⁸ : Algebra R S
p : Ideal R
inst✝¹⁷ : p.IsMaximal
Rₚ✝ : Type ?u.335608
Sₚ✝ : Type ?u.335611
inst✝¹⁶ : CommRing Rₚ✝
inst✝¹⁵ : CommRing Sₚ✝
inst✝¹⁴ : Algebra R Rₚ✝
inst✝¹³ : IsLocalization.AtPrime Rₚ✝ p
inst✝¹² : LocalRing Rₚ✝
inst✝¹¹ : CommRing Sₚ✝
inst✝¹⁰ : Algebra S Sₚ✝
inst✝⁹ : Algebra R Sₚ✝
inst✝⁸ : Algebra Rₚ✝ Sₚ✝
inst✝⁷ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ✝
inst✝⁶ : IsScalarTower R S Sₚ✝
inst✝⁵ : IsScalarTower R Rₚ✝ Sₚ✝
x : S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDomain S
inst✝² : NoZeroSMulDivisors R S
inst✝¹ : Module.Finite R S
inst✝ : IsIntegrallyClosed S
Rₚ : Type u_1 := Localization.AtPrime p
Sₚ : Type u_2 := Localization (algebraMapSubmonoid S p.primeCompl)
this✝³ : Algebra Rₚ Sₚ := localizationAlgebra p.primeCompl S
this✝² : IsScalarTower R Rₚ Sₚ
this✝¹ : IsLocalization (Submonoid.map (algebraMap R S) p.primeCompl) Sₚ
e : algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝ : IsDomain Sₚ
this : NoZeroSMulDivisors Rₚ Sₚ
⊢ (trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) = (Ideal.Quotient.mk p) ((intTrace R S) x)