Theorem dvdsmultr2d

Description: Deduction form of dvdsmultr2 . (Contributed by SN, 23-Aug-2024)

	Ref	Expression
Hypotheses	dvdsmultr2d.1	|- ( ph -> K e. ZZ )
	dvdsmultr2d.2	|- ( ph -> M e. ZZ )
	dvdsmultr2d.3	|- ( ph -> N e. ZZ )
	dvdsmultr2d.4	|- ( ph -> K || N )
Assertion	dvdsmultr2d	|- ( ph -> K || ( M x. N ) )

Proof

Step	Hyp	Ref	Expression
1		dvdsmultr2d.1	|- ( ph -> K e. ZZ )
2		dvdsmultr2d.2	|- ( ph -> M e. ZZ )
3		dvdsmultr2d.3	|- ( ph -> N e. ZZ )
4		dvdsmultr2d.4	|- ( ph -> K || N )
5		dvdsmultr2	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || N -> K || ( M x. N ) ) )
6	1 2 3 5	syl3anc	|- ( ph -> ( K || N -> K || ( M x. N ) ) )
7	4 6	mpd	|- ( ph -> K || ( M x. N ) )