Theorem coprmdvds2d

Description: If an integer is divisible by two coprime integers, then it is divisible by their product, a deduction version. (Contributed by metakunt, 12-May-2024)

	Ref	Expression
Hypotheses	coprmdvds2d.1	|- ( ph -> K e. ZZ )
	coprmdvds2d.2	|- ( ph -> M e. ZZ )
	coprmdvds2d.3	|- ( ph -> N e. ZZ )
	coprmdvds2d.4	|- ( ph -> ( K gcd M ) = 1 )
	coprmdvds2d.5	|- ( ph -> K || N )
	coprmdvds2d.6	|- ( ph -> M || N )
Assertion	coprmdvds2d	|- ( ph -> ( K x. M ) || N )

Proof

Step	Hyp	Ref	Expression
1		coprmdvds2d.1	|- ( ph -> K e. ZZ )
2		coprmdvds2d.2	|- ( ph -> M e. ZZ )
3		coprmdvds2d.3	|- ( ph -> N e. ZZ )
4		coprmdvds2d.4	|- ( ph -> ( K gcd M ) = 1 )
5		coprmdvds2d.5	|- ( ph -> K || N )
6		coprmdvds2d.6	|- ( ph -> M || N )
7	1 2 3	3jca	|- ( ph -> ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) )
8	7 4	jca	|- ( ph -> ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) )
9		coprmdvds2	|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( ( K || N /\ M || N ) -> ( K x. M ) || N ) )
10	8 9	syl	|- ( ph -> ( ( K || N /\ M || N ) -> ( K x. M ) || N ) )
11	5 6 10	mp2and	|- ( ph -> ( K x. M ) || N )