Step |
Hyp |
Ref |
Expression |
1 |
|
divides |
|- ( ( N e. ZZ /\ K e. ZZ ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) ) |
2 |
1
|
3adant1 |
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) ) |
3 |
2
|
adantr |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) ) |
4 |
|
simprr |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> x e. ZZ ) |
5 |
|
simpl2 |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> N e. ZZ ) |
6 |
|
zcn |
|- ( x e. ZZ -> x e. CC ) |
7 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
8 |
|
mulcom |
|- ( ( x e. CC /\ N e. CC ) -> ( x x. N ) = ( N x. x ) ) |
9 |
6 7 8
|
syl2an |
|- ( ( x e. ZZ /\ N e. ZZ ) -> ( x x. N ) = ( N x. x ) ) |
10 |
4 5 9
|
syl2anc |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( x x. N ) = ( N x. x ) ) |
11 |
10
|
breq2d |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || ( x x. N ) <-> M || ( N x. x ) ) ) |
12 |
|
simprl |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M gcd N ) = 1 ) |
13 |
|
simpl1 |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> M e. ZZ ) |
14 |
|
coprmdvds |
|- ( ( M e. ZZ /\ N e. ZZ /\ x e. ZZ ) -> ( ( M || ( N x. x ) /\ ( M gcd N ) = 1 ) -> M || x ) ) |
15 |
13 5 4 14
|
syl3anc |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( ( M || ( N x. x ) /\ ( M gcd N ) = 1 ) -> M || x ) ) |
16 |
12 15
|
mpan2d |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || ( N x. x ) -> M || x ) ) |
17 |
11 16
|
sylbid |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || ( x x. N ) -> M || x ) ) |
18 |
|
dvdsmulc |
|- ( ( M e. ZZ /\ x e. ZZ /\ N e. ZZ ) -> ( M || x -> ( M x. N ) || ( x x. N ) ) ) |
19 |
13 4 5 18
|
syl3anc |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || x -> ( M x. N ) || ( x x. N ) ) ) |
20 |
17 19
|
syld |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || ( x x. N ) -> ( M x. N ) || ( x x. N ) ) ) |
21 |
|
breq2 |
|- ( ( x x. N ) = K -> ( M || ( x x. N ) <-> M || K ) ) |
22 |
|
breq2 |
|- ( ( x x. N ) = K -> ( ( M x. N ) || ( x x. N ) <-> ( M x. N ) || K ) ) |
23 |
21 22
|
imbi12d |
|- ( ( x x. N ) = K -> ( ( M || ( x x. N ) -> ( M x. N ) || ( x x. N ) ) <-> ( M || K -> ( M x. N ) || K ) ) ) |
24 |
20 23
|
syl5ibcom |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( ( x x. N ) = K -> ( M || K -> ( M x. N ) || K ) ) ) |
25 |
24
|
anassrs |
|- ( ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) /\ x e. ZZ ) -> ( ( x x. N ) = K -> ( M || K -> ( M x. N ) || K ) ) ) |
26 |
25
|
rexlimdva |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( E. x e. ZZ ( x x. N ) = K -> ( M || K -> ( M x. N ) || K ) ) ) |
27 |
3 26
|
sylbid |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( N || K -> ( M || K -> ( M x. N ) || K ) ) ) |
28 |
27
|
impcomd |
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( M || K /\ N || K ) -> ( M x. N ) || K ) ) |