Theorem: ArithmeticFunction.IsMultiplicative.lcm_apply_mul_gcd_apply
File: Mathlib/NumberTheory/ArithmeticFunction.lean

tactic: by_cases hx : x = 0
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
state_after: case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : x = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

tactic: by_cases hy : y = 0
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
state_after: case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : y = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

tactic: have hgcd_ne_zero : x.gcd y ≠ 0 := gcd_ne_zero_left hx
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
state_after: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

tactic: have hlcm_ne_zero : x.lcm y ≠ 0 := lcm_ne_zero hx hy
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
state_after: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

tactic: have hfi_zero : ∀ {i}, f (i ^ 0) = 1 := by
  intro i; rw [Nat.pow_zero, hf.1]
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
state_after: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

tactic: iterate 4 rw [hf.multiplicative_factorization f (by assumption),
  Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
    (s := (x.primeFactors ⊔ y.primeFactors))]
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
state_after: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ (x.lcm y).factorization x_1)) *
      ∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ (x.gcd y).factorization x_1) =
    (∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ x.factorization x_1)) *
      ∏ x ∈ x.primeFactors ⊔ y.primeFactors, f (x ^ y.factorization x)

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ y.factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.gcd y).factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.lcm y).factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

tactic: simp only [hx, f.map_zero, zero_mul, Nat.lcm_zero_left, Nat.gcd_zero_left]
state_before: case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : x = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
state_after: no goals

tactic: simp only [hy, f.map_zero, mul_zero, Nat.lcm_zero_right, Nat.gcd_zero_right, zero_mul]
state_before: case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : y = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
state_after: no goals

tactic: intro i
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
⊢ ∀ {i : ℕ}, f (i ^ 0) = 1
state_after: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
i : ℕ
⊢ f (i ^ 0) = 1

tactic: rw [Nat.pow_zero, hf.1]
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
i : ℕ
⊢ f (i ^ 0) = 1
state_after: no goals

tactic: rw [hf.multiplicative_factorization f (by assumption),
Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
(s := (x.primeFactors ⊔ y.primeFactors))]
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ (x.lcm y).factorization x_1)) *
      ∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ (x.gcd y).factorization x_1) =
    (∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ x.factorization x_1)) * f y

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.gcd y).factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.lcm y).factorization.support ⊆ x.primeFactors ⊔ y.primeFactors
state_after: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ (x.lcm y).factorization x_1)) *
      ∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ (x.gcd y).factorization x_1) =
    (∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ x.factorization x_1)) *
      ∏ x ∈ x.primeFactors ⊔ y.primeFactors, f (x ^ y.factorization x)

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ y.factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.gcd y).factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.lcm y).factorization.support ⊆ x.primeFactors ⊔ y.primeFactors

tactic: assumption
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ y ≠ 0
state_after: no goals

tactic: rw [← Finset.prod_mul_distrib, ← Finset.prod_mul_distrib]
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ (x.lcm y).factorization x_1)) *
      ∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ (x.gcd y).factorization x_1) =
    (∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ x.factorization x_1)) *
      ∏ x ∈ x.primeFactors ⊔ y.primeFactors, f (x ^ y.factorization x)
state_after: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ ∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors,
      f (x_1 ^ (x.lcm y).factorization x_1) * f (x_1 ^ (x.gcd y).factorization x_1) =
    ∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ x.factorization x_1) * f (x_1 ^ y.factorization x_1)

tactic: apply Finset.prod_congr rfl
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ ∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors,
      f (x_1 ^ (x.lcm y).factorization x_1) * f (x_1 ^ (x.gcd y).factorization x_1) =
    ∏ x_1 ∈ x.primeFactors ⊔ y.primeFactors, f (x_1 ^ x.factorization x_1) * f (x_1 ^ y.factorization x_1)
state_after: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ ∀ x_1 ∈ x.primeFactors ⊔ y.primeFactors,
    f (x_1 ^ (x.lcm y).factorization x_1) * f (x_1 ^ (x.gcd y).factorization x_1) =
      f (x_1 ^ x.factorization x_1) * f (x_1 ^ y.factorization x_1)

tactic: intro p _
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ ∀ x_1 ∈ x.primeFactors ⊔ y.primeFactors,
    f (x_1 ^ (x.lcm y).factorization x_1) * f (x_1 ^ (x.gcd y).factorization x_1) =
      f (x_1 ^ x.factorization x_1) * f (x_1 ^ y.factorization x_1)
state_after: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
p : ℕ
a✝ : p ∈ x.primeFactors ⊔ y.primeFactors
⊢ f (p ^ (x.lcm y).factorization p) * f (p ^ (x.gcd y).factorization p) =
    f (p ^ x.factorization p) * f (p ^ y.factorization p)

tactic: rcases Nat.le_or_le (x.factorization p) (y.factorization p) with h | h <;>
  simp only [factorization_lcm hx hy, ge_iff_le, Finsupp.sup_apply, h, sup_of_le_right,
    sup_of_le_left, inf_of_le_right, Nat.factorization_gcd hx hy, Finsupp.inf_apply,
    inf_of_le_left, mul_comm]
state_before: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
p : ℕ
a✝ : p ∈ x.primeFactors ⊔ y.primeFactors
⊢ f (p ^ (x.lcm y).factorization p) * f (p ^ (x.gcd y).factorization p) =
    f (p ^ x.factorization p) * f (p ^ y.factorization p)
state_after: no goals

tactic: apply Finset.subset_union_right
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ y.factorization.support ⊆ x.primeFactors ⊔ y.primeFactors
state_after: no goals

tactic: apply Finset.subset_union_left
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.factorization.support ⊆ x.primeFactors ⊔ y.primeFactors
state_after: no goals

tactic: rw [factorization_gcd hx hy, Finsupp.support_inf, Finset.sup_eq_union]
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.gcd y).factorization.support ⊆ x.primeFactors ⊔ y.primeFactors
state_after: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.factorization.support ∩ y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors

tactic: apply Finset.inter_subset_union
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.factorization.support ∩ y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
state_after: no goals

tactic: simp [factorization_lcm hx hy]
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.lcm y).factorization.support ⊆ x.primeFactors ⊔ y.primeFactors
state_after: no goals