Theorem dvdsmultr1d

Description: Deduction form of dvdsmultr1 . (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	dvdsmultr1d.1	|- ( ph -> K e. ZZ )
	dvdsmultr1d.2	|- ( ph -> M e. ZZ )
	dvdsmultr1d.3	|- ( ph -> N e. ZZ )
	dvdsmultr1d.4	|- ( ph -> K || M )
Assertion	dvdsmultr1d	|- ( ph -> K || ( M x. N ) )

Proof

Step	Hyp	Ref	Expression
1		dvdsmultr1d.1	|- ( ph -> K e. ZZ )
2		dvdsmultr1d.2	|- ( ph -> M e. ZZ )
3		dvdsmultr1d.3	|- ( ph -> N e. ZZ )
4		dvdsmultr1d.4	|- ( ph -> K || M )
5		dvdsmultr1	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || M -> K || ( M x. N ) ) )
6	1 2 3 5	syl3anc	|- ( ph -> ( K || M -> K || ( M x. N ) ) )
7	4 6	mpd	|- ( ph -> K || ( M x. N ) )