Step |
Hyp |
Ref |
Expression |
1 |
|
lcmcom |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 lcm 𝑁 ) = ( 𝑁 lcm 𝑀 ) ) |
2 |
1
|
adantr |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 = 0 ) → ( 𝑀 lcm 𝑁 ) = ( 𝑁 lcm 𝑀 ) ) |
3 |
|
oveq2 |
⊢ ( 𝑀 = 0 → ( 𝑁 lcm 𝑀 ) = ( 𝑁 lcm 0 ) ) |
4 |
|
lcm0val |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 lcm 0 ) = 0 ) |
5 |
3 4
|
sylan9eqr |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑀 = 0 ) → ( 𝑁 lcm 𝑀 ) = 0 ) |
6 |
5
|
adantll |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 = 0 ) → ( 𝑁 lcm 𝑀 ) = 0 ) |
7 |
2 6
|
eqtrd |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 = 0 ) → ( 𝑀 lcm 𝑁 ) = 0 ) |
8 |
|
oveq2 |
⊢ ( 𝑁 = 0 → ( 𝑀 lcm 𝑁 ) = ( 𝑀 lcm 0 ) ) |
9 |
|
lcm0val |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 lcm 0 ) = 0 ) |
10 |
8 9
|
sylan9eqr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 = 0 ) → ( 𝑀 lcm 𝑁 ) = 0 ) |
11 |
10
|
adantlr |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 = 0 ) → ( 𝑀 lcm 𝑁 ) = 0 ) |
12 |
7 11
|
jaodan |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑀 = 0 ∨ 𝑁 = 0 ) ) → ( 𝑀 lcm 𝑁 ) = 0 ) |
13 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
14 |
12 13
|
eqeltrdi |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑀 = 0 ∨ 𝑁 = 0 ) ) → ( 𝑀 lcm 𝑁 ) ∈ ℕ0 ) |
15 |
|
lcmn0cl |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∨ 𝑁 = 0 ) ) → ( 𝑀 lcm 𝑁 ) ∈ ℕ ) |
16 |
15
|
nnnn0d |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∨ 𝑁 = 0 ) ) → ( 𝑀 lcm 𝑁 ) ∈ ℕ0 ) |
17 |
14 16
|
pm2.61dan |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 lcm 𝑁 ) ∈ ℕ0 ) |