Step |
Hyp |
Ref |
Expression |
1 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
2 |
|
zcn |
|- ( M e. ZZ -> M e. CC ) |
3 |
|
mulcom |
|- ( ( N e. CC /\ M e. CC ) -> ( N x. M ) = ( M x. N ) ) |
4 |
1 2 3
|
syl2anr |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N x. M ) = ( M x. N ) ) |
5 |
|
zmulcl |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ ) |
6 |
|
dvds0lem |
|- ( ( ( N e. ZZ /\ M e. ZZ /\ ( M x. N ) e. ZZ ) /\ ( N x. M ) = ( M x. N ) ) -> M || ( M x. N ) ) |
7 |
6
|
ex |
|- ( ( N e. ZZ /\ M e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( N x. M ) = ( M x. N ) -> M || ( M x. N ) ) ) |
8 |
7
|
3com12 |
|- ( ( M e. ZZ /\ N e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( N x. M ) = ( M x. N ) -> M || ( M x. N ) ) ) |
9 |
5 8
|
mpd3an3 |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( N x. M ) = ( M x. N ) -> M || ( M x. N ) ) ) |
10 |
4 9
|
mpd |
|- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) ) |