Theorem muldvds2d

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	muldvds2d.1	|- ( ph -> K e. ZZ )
	muldvds2d.2	|- ( ph -> M e. ZZ )
	muldvds2d.3	|- ( ph -> N e. ZZ )
	muldvds2d.4	|- ( ph -> ( K x. M ) || N )
Assertion	muldvds2d	|- ( ph -> M || N )

Proof

Step	Hyp	Ref	Expression
1		muldvds2d.1	|- ( ph -> K e. ZZ )
2		muldvds2d.2	|- ( ph -> M e. ZZ )
3		muldvds2d.3	|- ( ph -> N e. ZZ )
4		muldvds2d.4	|- ( ph -> ( K x. M ) || N )
5	1 2 3	3jca	|- ( ph -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
6		muldvds2	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> M || N ) )
7	5 4 6	sylc	|- ( ph -> M || N )