-- Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicative.lean
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}, IsRelPrime x y → f (x * y) = f x * f y) :
    f (a * b) = f a * f b

Proof state:
case intro
α : Type u_1
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}, IsRelPrime x y → f (x * y) = f x * f y
this✝² : DecidableEq α := Classical.decEq α
ha0 : ¬a = 0
hb0 : ¬b = 0
hf1 : ¬f 1 = 0
this✝¹ : Nontrivial α
this✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
this :
  f
      (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =
    f
        (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
          p ^ Multiset.count p (normalizedFactors a)) *
      f
        (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
          p ^ Multiset.count p (normalizedFactors b))
ua : αˣ
a_eq : (normalizedFactors a).prod * ↑ua = a
⊢ f (a * b) = f a * f b

Tactic:
obtain ⟨ub, b_eq⟩ := prod_normalizedFactors hb0

Result:
case intro.intro
α : Type u_1
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}, IsRelPrime x y → f (x * y) = f x * f y
this✝² : DecidableEq α := Classical.decEq α
ha0 : ¬a = 0
hb0 : ¬b = 0
hf1 : ¬f 1 = 0
this✝¹ : Nontrivial α
this✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
this :
  f
      (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =
    f
        (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
          p ^ Multiset.count p (normalizedFactors a)) *
      f
        (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
          p ^ Multiset.count p (normalizedFactors b))
ua : αˣ
a_eq : (normalizedFactors a).prod * ↑ua = a
ub : αˣ
b_eq : (normalizedFactors b).prod * ↑ub = b
⊢ f (a * b) = f a * f b

Proof state:
case neg.refine_2
α : Type u_1
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}, IsRelPrime x y → 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
⊢ ∀ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
    ∀ q ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset, p ∣ q → p = q

Tactic:
simp only [Multiset.mem_toFinset, Finset.mem_union]

Result:
case neg.refine_2
α : Type u_1
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}, IsRelPrime x y → 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
⊢ ∀ (p : α),
    p ∈ normalizedFactors a ∨ p ∈ normalizedFactors b →
      ∀ (q : α), q ∈ normalizedFactors a ∨ q ∈ normalizedFactors b → p ∣ q → p = q