have H : RingHom.comp (algebraMap (FractionRing R) (FractionRing S))
  ↑(FractionRing.algEquiv R K).symm.toRingEquiv =
    RingHom.comp ↑(FractionRing.algEquiv S L).symm.toRingEquiv (algebraMap K L) := by
  apply IsLocalization.ringHom_ext R⁰
  ext
  simp only [AlgEquiv.toRingEquiv_eq_coe, RingHom.coe_comp, RingHom.coe_coe,
    AlgEquiv.coe_ringEquiv, Function.comp_apply, AlgEquiv.commutes,
    ← IsScalarTower.algebraMap_apply]
  rw [IsScalarTower.algebraMap_apply R S L, AlgEquiv.commutes, ← IsScalarTower.algebraMap_apply]
R : Type u_1
K : Type u_2
L : Type u_3
S : Type u_4
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Algebra R K
inst✝⁸ : IsFractionRing R K
inst✝⁷ : Algebra S L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : IsScalarTower R S L
inst✝³ : IsScalarTower R K L
inst✝² : IsIntegralClosure S R L
inst✝¹ : FiniteDimensional K L
inst✝ : IsSeparable K L
p : Ideal R
hp : p.IsPrime
hpbot : p ≠ ⊥
x : S
hx' : Algebra.adjoin K {(algebraMap S L) x} = ⊤
h : (map (Ideal.Quotient.mk p) (minpoly R x)).Separable
this✝⁴ : IsDomain S
hRS : Function.Injective ⇑(algebraMap R S)
this✝³ : NoZeroSMulDivisors R S
this✝² : IsNoetherian R S
this✝¹ : IsDedekindDomain S
this✝ : IsFractionRing S L
this : (aeval x) (derivative (minpoly R x)) ∈ differentIdeal R S
⊢ IsUnramifiedAt S p
R : Type u_1
K : Type u_2
L : Type u_3
S : Type u_4
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Algebra R K
inst✝⁸ : IsFractionRing R K
inst✝⁷ : Algebra S L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : IsScalarTower R S L
inst✝³ : IsScalarTower R K L
inst✝² : IsIntegralClosure S R L
inst✝¹ : FiniteDimensional K L
inst✝ : IsSeparable K L
p : Ideal R
hp : p.IsPrime
hpbot : p ≠ ⊥
x : S
hx' : Algebra.adjoin K {(algebraMap S L) x} = ⊤
h : (map (Ideal.Quotient.mk p) (minpoly R x)).Separable
this✝⁴ : IsDomain S
hRS : Function.Injective ⇑(algebraMap R S)
this✝³ : NoZeroSMulDivisors R S
this✝² : IsNoetherian R S
this✝¹ : IsDedekindDomain S
this✝ : IsFractionRing S L
this : (aeval x) (derivative (minpoly R x)) ∈ differentIdeal R S
H :
  (algebraMap (FractionRing R) (FractionRing S)).comp ↑(FractionRing.algEquiv R K).symm.toRingEquiv =
    (↑(FractionRing.algEquiv S L).symm.toRingEquiv).comp (algebraMap K L)
⊢ IsUnramifiedAt S p