Theorem gcdmultipled

Description: The greatest common divisor of a nonnegative integer M and a multiple of it is M itself. (Contributed by Rohan Ridenour, 3-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		gcdmultipled.1	|- ( ph -> M e. NN0 )
2		gcdmultipled.2	|- ( ph -> N e. ZZ )
3	1	nn0zd	|- ( ph -> M e. ZZ )
4		0zd	|- ( ph -> 0 e. ZZ )
5		gcdaddm	|- ( ( N e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) -> ( M gcd 0 ) = ( M gcd ( 0 + ( N x. M ) ) ) )
6	2 3 4 5	syl3anc	|- ( ph -> ( M gcd 0 ) = ( M gcd ( 0 + ( N x. M ) ) ) )
7		nn0gcdid0	|- ( M e. NN0 -> ( M gcd 0 ) = M )
8	1 7	syl	|- ( ph -> ( M gcd 0 ) = M )
9	2 3	zmulcld	|- ( ph -> ( N x. M ) e. ZZ )
10	9	zcnd	|- ( ph -> ( N x. M ) e. CC )
11	10	addid2d	|- ( ph -> ( 0 + ( N x. M ) ) = ( N x. M ) )
12	11	oveq2d	|- ( ph -> ( M gcd ( 0 + ( N x. M ) ) ) = ( M gcd ( N x. M ) ) )
13	6 8 12	3eqtr3rd	|- ( ph -> ( M gcd ( N x. M ) ) = M )