Theorem muldvds1d

Description: If a product divides an integer, so does one of its factors, a deduction version. (Contributed by metakunt, 12-May-2024)

	Ref	Expression
Hypotheses	muldvds1d.1	|- ( ph -> K e. ZZ )
	muldvds1d.2	|- ( ph -> M e. ZZ )
	muldvds1d.3	|- ( ph -> N e. ZZ )
	muldvds1d.4	|- ( ph -> ( K x. M ) || N )
Assertion	muldvds1d	|- ( ph -> K || N )

Proof

Step	Hyp	Ref	Expression
1		muldvds1d.1	|- ( ph -> K e. ZZ )
2		muldvds1d.2	|- ( ph -> M e. ZZ )
3		muldvds1d.3	|- ( ph -> N e. ZZ )
4		muldvds1d.4	|- ( ph -> ( K x. M ) || N )
5	1 2 3	3jca	|- ( ph -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
6		muldvds1	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> K || N ) )
7	5 6	syl	|- ( ph -> ( ( K x. M ) || N -> K || N ) )
8	4 7	mpd	|- ( ph -> K || N )