Step |
Hyp |
Ref |
Expression |
1 |
|
coprmdvds2d.1 |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
2 |
|
coprmdvds2d.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
coprmdvds2d.3 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
4 |
|
coprmdvds2d.4 |
⊢ ( 𝜑 → ( 𝐾 gcd 𝑀 ) = 1 ) |
5 |
|
coprmdvds2d.5 |
⊢ ( 𝜑 → 𝐾 ∥ 𝑁 ) |
6 |
|
coprmdvds2d.6 |
⊢ ( 𝜑 → 𝑀 ∥ 𝑁 ) |
7 |
1 2 3
|
3jca |
⊢ ( 𝜑 → ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) |
8 |
7 4
|
jca |
⊢ ( 𝜑 → ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) ) |
9 |
|
coprmdvds2 |
⊢ ( ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) → ( ( 𝐾 ∥ 𝑁 ∧ 𝑀 ∥ 𝑁 ) → ( 𝐾 · 𝑀 ) ∥ 𝑁 ) ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → ( ( 𝐾 ∥ 𝑁 ∧ 𝑀 ∥ 𝑁 ) → ( 𝐾 · 𝑀 ) ∥ 𝑁 ) ) |
11 |
5 6 10
|
mp2and |
⊢ ( 𝜑 → ( 𝐾 · 𝑀 ) ∥ 𝑁 ) |