Theorem dvds2addd

Description: Deduction form of dvds2add . (Contributed by SN, 21-Aug-2024)

	Ref	Expression
Hypotheses	dvds2addd.k	|- ( ph -> K e. ZZ )
	dvds2addd.m	|- ( ph -> M e. ZZ )
	dvds2addd.n	|- ( ph -> N e. ZZ )
	dvds2addd.1	|- ( ph -> K || M )
	dvds2addd.2	|- ( ph -> K || N )
Assertion	dvds2addd	|- ( ph -> K || ( M + N ) )

Proof

Step	Hyp	Ref	Expression
1		dvds2addd.k	|- ( ph -> K e. ZZ )
2		dvds2addd.m	|- ( ph -> M e. ZZ )
3		dvds2addd.n	|- ( ph -> N e. ZZ )
4		dvds2addd.1	|- ( ph -> K || M )
5		dvds2addd.2	|- ( ph -> K || N )
6		dvds2add	|- ( ( 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 ) )