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 )