Theorem dvdstrd

Description: The divides relation is transitive, a deduction version of dvdstr . (Contributed by metakunt, 12-May-2024)

	Ref	Expression
Hypotheses	dvdstrd.1	|- ( ph -> K e. ZZ )
	dvdstrd.2	|- ( ph -> M e. ZZ )
	dvdstrd.3	|- ( ph -> N e. ZZ )
	dvdstrd.4	|- ( ph -> K || M )
	dvdstrd.5	|- ( ph -> M || N )
Assertion	dvdstrd	|- ( ph -> K || N )

Proof

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