-- Module: Mathlib.RingTheory.Conductor
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
x : S
I : Ideal R
hx : comap (F := R →+* S) (algebraMap R S) (conductor R x) ⊔ I = ⊤
h_alg : Function.Injective (α := ↥R[x]) (β := S) (⇑(algebraMap (↥R[x]) S) : (a : ↥R[x]) → S)
f : HasQuotient.Quotient ↥R[x] (B := Ideal ↥R[x])
    (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I) →+*
  HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I) :=
  Ideal.Quotient.lift (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I)
    (RingHom.comp (Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) I)) (algebraMap (↥R[x]) S)) ⋯
u : ↥R[x]
hu :
  Membership.mem (γ := Submodule ↥R[x] ↥R[x])
    (ker (F := ↥R[x] →+* HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I))
      (RingHom.comp (Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) I)) (algebraMap (↥R[x]) S)))
    u
refine_1 :
  ∀
    (hu :
      Eq (α := HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I))
        ((RingHom.comp (Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) I)) (algebraMap (↥R[x]) S) :
              (a : ↥R[x]) → S ⧸ Ideal.map (algebraMap R S) I)
            u :
          HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I))
        (0 : HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I))),
    Membership.mem (γ := Submodule ↥R[x] ↥R[x]) (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I) u
refine_1₁ :
  ∀
    (hu :
      Eq (α := HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I))
        ((Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) I) : (a : S) → S ⧸ Ideal.map (algebraMap R S) I)
            ((algebraMap (↥R[x]) S : (a : ↥R[x]) → S) u) :
          HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I))
        (0 : HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I))),
    Membership.mem (γ := Submodule ↥R[x] ↥R[x]) (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I) u
refine_1₂ :
  ∀
    (hu :
      Membership.mem (α := S) (γ := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I)
        ((algebraMap (↥R[x]) S : (a : ↥R[x]) → S) u : S)),
    Membership.mem (γ := Submodule ↥R[x] ↥R[x]) (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I) u
refine_1₃ :
  ∀
    (hu :
      Membership.mem (γ := Ideal ↥R[x])
        (comap (F := ↥R[x] →+* S) (algebraMap (↥R[x]) S) (Ideal.map (F := R →+* S) (algebraMap R S) I)) u),
    Membership.mem (γ := Submodule ↥R[x] ↥R[x]) (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I) u
refine_1₄ refine_1₅ refine_1₆ refine_1₇ :
  ∀ (hu : Membership.mem (γ := Ideal ↥R[x]) (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I) u),
    Membership.mem (γ := Submodule ↥R[x] ↥R[x]) (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I) u
⊢ Membership.mem (γ := Submodule ↥R[x] ↥R[x]) (Ideal.map (F := RingHom.{u_1, u_2} R ↥R[x]) (algebraMap R ↥R[x]) I) u

Tactic:
rwa [RingHom.mem_ker, RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, ← Ideal.mem_comap,
      comap_map_eq_map_adjoin_of_coprime_conductor hx h_alg] at hu

Premises used: RingHom.mem_ker, Ideal.Quotient.eq_zero_iff_mem, RingHom.comp_apply, Ideal.mem_comap, comap_map_eq_map_adjoin_of_coprime_conductor