Step |
Hyp |
Ref |
Expression |
1 |
|
odcl.1 |
|- X = ( Base ` G ) |
2 |
|
odcl.2 |
|- O = ( od ` G ) |
3 |
|
odid.3 |
|- .x. = ( .g ` G ) |
4 |
|
odid.4 |
|- .0. = ( 0g ` G ) |
5 |
|
zsubcl |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ ) |
6 |
1 2 3 4
|
oddvds |
|- ( ( G e. Grp /\ A e. X /\ ( M - N ) e. ZZ ) -> ( ( O ` A ) || ( M - N ) <-> ( ( M - N ) .x. A ) = .0. ) ) |
7 |
5 6
|
syl3an3 |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( O ` A ) || ( M - N ) <-> ( ( M - N ) .x. A ) = .0. ) ) |
8 |
|
simp1 |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> G e. Grp ) |
9 |
|
simp3l |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> M e. ZZ ) |
10 |
|
simp3r |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> N e. ZZ ) |
11 |
|
simp2 |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> A e. X ) |
12 |
|
eqid |
|- ( -g ` G ) = ( -g ` G ) |
13 |
1 3 12
|
mulgsubdir |
|- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ A e. X ) ) -> ( ( M - N ) .x. A ) = ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) ) |
14 |
8 9 10 11 13
|
syl13anc |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( M - N ) .x. A ) = ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) ) |
15 |
14
|
eqeq1d |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( ( M - N ) .x. A ) = .0. <-> ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) = .0. ) ) |
16 |
1 3
|
mulgcl |
|- ( ( G e. Grp /\ M e. ZZ /\ A e. X ) -> ( M .x. A ) e. X ) |
17 |
8 9 11 16
|
syl3anc |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M .x. A ) e. X ) |
18 |
1 3
|
mulgcl |
|- ( ( G e. Grp /\ N e. ZZ /\ A e. X ) -> ( N .x. A ) e. X ) |
19 |
8 10 11 18
|
syl3anc |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N .x. A ) e. X ) |
20 |
1 4 12
|
grpsubeq0 |
|- ( ( G e. Grp /\ ( M .x. A ) e. X /\ ( N .x. A ) e. X ) -> ( ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) = .0. <-> ( M .x. A ) = ( N .x. A ) ) ) |
21 |
8 17 19 20
|
syl3anc |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) = .0. <-> ( M .x. A ) = ( N .x. A ) ) ) |
22 |
7 15 21
|
3bitrd |
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( O ` A ) || ( M - N ) <-> ( M .x. A ) = ( N .x. A ) ) ) |