/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
  obtain ⟨l, H, H'⟩ := hz'
  rw [Finsupp.total_apply] at H'
  rw [← H', mul_comm, Finsupp.sum_mul]
  have lem : ∀ {a : R}, a ∈ I → l a • algebraMap R S a * algebraMap R S p ∈
      algebraMap R<x> S '' I.map (algebraMap R R<x>) := by
    intro a ha
    rw [Algebra.id.smul_eq_mul, mul_assoc, mul_comm, mul_assoc, Set.mem_image]
    refine Exists.intro
        (algebraMap R R<x> a * ⟨l a * algebraMap R S p,
          show l a * algebraMap R S p ∈ R<x> from ?h⟩) ?_
    case h =>
      rw [mul_comm]
      exact mem_conductor_iff.mp (Ideal.mem_comap.mp hp) _
    · refine ⟨?_, ?_⟩
      · rw [mul_comm]
        apply Ideal.mul_mem_left (I.map (algebraMap R R<x>)) _ (Ideal.mem_map_of_mem _ ha)
      · simp only [RingHom.map_mul, mul_comm (algebraMap R S p) (l a)]
        rfl
  refine Finset.sum_induction _ (fun u => u ∈ algebraMap R<x> S '' I.map (algebraMap R R<x>))
      (fun a b => ?_) ?_ ?_
  rintro ⟨z, hz, rfl⟩ ⟨y, hy, rfl⟩
  rw [← RingHom.map_add]
  

Proof state:
case intro.intro.refine_1.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z✝ : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.sum l fun i a => a • (algebraMap R S) i) = z✝
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈
        ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
z : ↥(adjoin R {x})
hz : z ∈ ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
y : ↥(adjoin R {x})
hy : y ∈ ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
⊢ (algebraMap (↥(adjoin R {x})) S) (z + y) ∈
    ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
case intro.intro.refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.sum l fun i a => a • (algebraMap R S) i) = z
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈
        ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
⊢ (fun u => u ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)) 0
case intro.intro.refine_3
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.sum l fun i a => a • (algebraMap R S) i) = z
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈
        ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
⊢ ∀ x_1 ∈ l.support,
    (fun u => u ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I))
      ((fun a c => c • (algebraMap R S) a * (algebraMap R S) p) x_1 (l x_1))

Next tactic:
exact ⟨z + y, Ideal.add_mem _ (SetLike.mem_coe.mp hz) hy, rfl⟩