Metamath Proof Explorer


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 )