Metamath Proof Explorer
Description: Closure of the gcd operator. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
gcdcld.1 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
|
|
gcdcld.2 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
|
Assertion |
gcdcld |
⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) ∈ ℕ0 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
gcdcld.1 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
gcdcld.2 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
3 |
|
gcdcl |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ0 ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) ∈ ℕ0 ) |