obtain ⟨c, rfl⟩ : a ∣ b := by
  refine' IH ha.2 (fun p hp ↦ (le_trans ?_ (H p hp)))
  rw [multiplicity.mul hp]
  exact le_add_self
case neg.hi
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝⁵ : Field K
inst✝⁴ : CharZero K
inst✝³ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : DecidableRel fun a b => a ∣ b
inst✝ : UniqueFactorizationMonoid M
b : M
hb : ¬b = 0
a q : M
a✝ : a ≠ 0
hq : Prime q
IH : a ≠ 0 → (∀ (p : M), Prime p → multiplicity p a ≤ multiplicity p b) → a ∣ b
H : ∀ (p : M), Prime p → multiplicity p (q * a) ≤ multiplicity p b
ha : ¬q = 0 ∧ ¬a = 0
⊢ q * a ∣ b
case neg.hi.intro
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝⁵ : Field K
inst✝⁴ : CharZero K
inst✝³ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : DecidableRel fun a b => a ∣ b
inst✝ : UniqueFactorizationMonoid M
a q : M
a✝ : a ≠ 0
hq : Prime q
ha : ¬q = 0 ∧ ¬a = 0
c : M
hb : ¬a * c = 0
IH : a ≠ 0 → (∀ (p : M), Prime p → multiplicity p a ≤ multiplicity p (a * c)) → a ∣ a * c
H : ∀ (p : M), Prime p → multiplicity p (q * a) ≤ multiplicity p (a * c)
⊢ q * a ∣ a * c