/-- If `f` maps `p ^ i` to `(f p) ^ i` for primes `p`, and `f`
is multiplicative on coprime elements, then `f` is multiplicative everywhere. -/
theorem multiplicative_of_coprime (f : α → β) (a b : α) (h0 : f 0 = 0)
    (h1 : ∀ {x y}, IsUnit y → f (x * y) = f x * f y)
    (hpr : ∀ {p} (i : ℕ), Prime p → f (p ^ i) = f p ^ i)
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y) :
    f (a * b) = f a * f b := by
  letI := Classical.decEq α
  by_cases ha0 : a = 0
  · rw [ha0, zero_mul, h0, zero_mul]
  by_cases hb0 : b = 0
  · rw [hb0, mul_zero, h0, mul_zero]
  by_cases hf1 : f 1 = 0
  · calc
      f (a * b) = f (a * b * 1) := by rw [mul_one]
      _ = 0 := by simp only [h1 isUnit_one, hf1, mul_zero]
      _ = f a * f (b * 1) := by simp only [h1 isUnit_one, hf1, mul_zero]
      _ = f a * f b := by rw [mul_one]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
  suffices
      f (∏ p in (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
        p ^ ((normalizedFactors a).count p + (normalizedFactors b).count p)) =
      f (∏ p in (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
        p ^ (normalizedFactors a).count p) *
      f (∏ p : α in (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
        p ^ (normalizedFactors b).count p) by
    obtain ⟨ua, a_eq⟩ := normalizedFactors_prod ha0
    obtain ⟨ub, b_eq⟩ := normalizedFactors_prod hb0
    rw [← a_eq, ← b_eq, mul_right_comm (Multiset.prod (normalizedFactors a)) ua
        (Multiset.prod (normalizedFactors b) * ub), h1 ua.isUnit, h1 ub.isUnit, h1 ua.isUnit, ←
      mul_assoc, h1 ub.isUnit, mul_right_comm _ (f ua), ← mul_assoc]
    congr
    rw [← (normalizedFactors a).map_id, ← (normalizedFactors b).map_id,
      Finset.prod_multiset_map_count, Finset.prod_multiset_map_count,
      Finset.prod_subset (Finset.subset_union_left _ (normalizedFactors b).toFinset),
      Finset.prod_subset (Finset.subset_union_right _ (normalizedFactors b).toFinset), ←
      Finset.prod_mul_distrib]
    simp_rw [id.def, ← pow_add, this]
    all_goals simp only [Multiset.mem_toFinset]
    · intro p _ hpb
      simp [hpb]
    · intro p _ hpa
      simp [hpa]
  

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
f : α → β
a b : α
h0 : f 0 = 0
h1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y
hpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y
this✝¹ : DecidableEq α := Classical.decEq α
ha0 : ¬a = 0
hb0 : ¬b = 0
hf1 : ¬f 1 = 0
this✝ : Nontrivial α
this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
⊢ f
      (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),
        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =
    f
        (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),
          p ^ Multiset.count p (normalizedFactors a)) *
      f
        (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),
          p ^ Multiset.count p (normalizedFactors b))

Next tactic:
refine' multiplicative_prime_power _ _ _ _ _ @h1 @hpr @hcp