Theorem gcdcld

Description: Closure of the gcd operator. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	gcdcld.1	|- ( ph -> M e. ZZ )
	gcdcld.2	|- ( ph -> N e. ZZ )
Assertion	gcdcld	|- ( ph -> ( M gcd N ) e. NN0 )

Proof

Step	Hyp	Ref	Expression
1		gcdcld.1	|- ( ph -> M e. ZZ )
2		gcdcld.2	|- ( ph -> N e. ZZ )
3		gcdcl	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
4	1 2 3	syl2anc	|- ( ph -> ( M gcd N ) e. NN0 )