Theorem gcdmultiplei

Description: The GCD of a multiple of a positive integer is the positive integer itself. (Contributed by metakunt, 25-Apr-2024)

	Ref	Expression
Hypotheses	gcdmultiplei.1	|- M e. NN
	gcdmultiplei.2	|- N e. NN
Assertion	gcdmultiplei	|- ( M gcd ( M x. N ) ) = M

Proof

Step	Hyp	Ref	Expression
1		gcdmultiplei.1	|- M e. NN
2		gcdmultiplei.2	|- N e. NN
3		gcdmultiple	|- ( ( M e. NN /\ N e. NN ) -> ( M gcd ( M x. N ) ) = M )
4	1 2 3	mp2an	|- ( M gcd ( M x. N ) ) = M