Theorem dvds2subd

Description: Deduction form of dvds2sub . (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	dvds2subd.k	|- ( ph -> K e. ZZ )
	dvds2subd.m	|- ( ph -> M e. ZZ )
	dvds2subd.n	|- ( ph -> N e. ZZ )
	dvds2subd.1	|- ( ph -> K || M )
	dvds2subd.2	|- ( ph -> K || N )
Assertion	dvds2subd	|- ( ph -> K || ( M - N ) )

Proof

Step	Hyp	Ref	Expression
1		dvds2subd.k	|- ( ph -> K e. ZZ )
2		dvds2subd.m	|- ( ph -> M e. ZZ )
3		dvds2subd.n	|- ( ph -> N e. ZZ )
4		dvds2subd.1	|- ( ph -> K || M )
5		dvds2subd.2	|- ( ph -> K || N )
6		dvds2sub	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M - N ) ) )
7	1 2 3 6	syl3anc	|- ( ph -> ( ( K || M /\ K || N ) -> K || ( M - N ) ) )
8	4 5 7	mp2and	|- ( ph -> K || ( M - N ) )