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Metamath Proof Explorer

Theorem gcdcld

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 )