Step |
Hyp |
Ref |
Expression |
1 |
|
nnne0 |
|- ( M e. NN -> M =/= 0 ) |
2 |
1
|
adantr |
|- ( ( M e. NN /\ N e. ZZ ) -> M =/= 0 ) |
3 |
|
nncn |
|- ( M e. NN -> M e. CC ) |
4 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
5 |
|
zcn |
|- ( k e. ZZ -> k e. CC ) |
6 |
|
divcan3 |
|- ( ( k e. CC /\ M e. CC /\ M =/= 0 ) -> ( ( M x. k ) / M ) = k ) |
7 |
6
|
3coml |
|- ( ( M e. CC /\ M =/= 0 /\ k e. CC ) -> ( ( M x. k ) / M ) = k ) |
8 |
7
|
3expa |
|- ( ( ( M e. CC /\ M =/= 0 ) /\ k e. CC ) -> ( ( M x. k ) / M ) = k ) |
9 |
5 8
|
sylan2 |
|- ( ( ( M e. CC /\ M =/= 0 ) /\ k e. ZZ ) -> ( ( M x. k ) / M ) = k ) |
10 |
9
|
3adantl2 |
|- ( ( ( M e. CC /\ N e. CC /\ M =/= 0 ) /\ k e. ZZ ) -> ( ( M x. k ) / M ) = k ) |
11 |
|
oveq1 |
|- ( ( M x. k ) = N -> ( ( M x. k ) / M ) = ( N / M ) ) |
12 |
10 11
|
sylan9req |
|- ( ( ( ( M e. CC /\ N e. CC /\ M =/= 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k = ( N / M ) ) |
13 |
|
simplr |
|- ( ( ( ( M e. CC /\ N e. CC /\ M =/= 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k e. ZZ ) |
14 |
12 13
|
eqeltrrd |
|- ( ( ( ( M e. CC /\ N e. CC /\ M =/= 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> ( N / M ) e. ZZ ) |
15 |
14
|
rexlimdva2 |
|- ( ( M e. CC /\ N e. CC /\ M =/= 0 ) -> ( E. k e. ZZ ( M x. k ) = N -> ( N / M ) e. ZZ ) ) |
16 |
|
divcan2 |
|- ( ( N e. CC /\ M e. CC /\ M =/= 0 ) -> ( M x. ( N / M ) ) = N ) |
17 |
16
|
3com12 |
|- ( ( M e. CC /\ N e. CC /\ M =/= 0 ) -> ( M x. ( N / M ) ) = N ) |
18 |
|
oveq2 |
|- ( k = ( N / M ) -> ( M x. k ) = ( M x. ( N / M ) ) ) |
19 |
18
|
eqeq1d |
|- ( k = ( N / M ) -> ( ( M x. k ) = N <-> ( M x. ( N / M ) ) = N ) ) |
20 |
19
|
rspcev |
|- ( ( ( N / M ) e. ZZ /\ ( M x. ( N / M ) ) = N ) -> E. k e. ZZ ( M x. k ) = N ) |
21 |
20
|
expcom |
|- ( ( M x. ( N / M ) ) = N -> ( ( N / M ) e. ZZ -> E. k e. ZZ ( M x. k ) = N ) ) |
22 |
17 21
|
syl |
|- ( ( M e. CC /\ N e. CC /\ M =/= 0 ) -> ( ( N / M ) e. ZZ -> E. k e. ZZ ( M x. k ) = N ) ) |
23 |
15 22
|
impbid |
|- ( ( M e. CC /\ N e. CC /\ M =/= 0 ) -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) ) |
24 |
23
|
3expia |
|- ( ( M e. CC /\ N e. CC ) -> ( M =/= 0 -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) ) ) |
25 |
3 4 24
|
syl2an |
|- ( ( M e. NN /\ N e. ZZ ) -> ( M =/= 0 -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) ) ) |
26 |
2 25
|
mpd |
|- ( ( M e. NN /\ N e. ZZ ) -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) ) |