Step |
Hyp |
Ref |
Expression |
1 |
|
0nsg.z |
|- .0. = ( 0g ` G ) |
2 |
1
|
0subg |
|- ( G e. Grp -> { .0. } e. ( SubGrp ` G ) ) |
3 |
|
elsni |
|- ( y e. { .0. } -> y = .0. ) |
4 |
3
|
ad2antll |
|- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y = .0. ) |
5 |
4
|
oveq2d |
|- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` G ) .0. ) ) |
6 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
7 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
8 |
6 7 1
|
grprid |
|- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( x ( +g ` G ) .0. ) = x ) |
9 |
8
|
adantrr |
|- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) .0. ) = x ) |
10 |
5 9
|
eqtrd |
|- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = x ) |
11 |
10
|
oveq1d |
|- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = ( x ( -g ` G ) x ) ) |
12 |
|
eqid |
|- ( -g ` G ) = ( -g ` G ) |
13 |
6 1 12
|
grpsubid |
|- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( x ( -g ` G ) x ) = .0. ) |
14 |
13
|
adantrr |
|- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( -g ` G ) x ) = .0. ) |
15 |
11 14
|
eqtrd |
|- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = .0. ) |
16 |
|
ovex |
|- ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. _V |
17 |
16
|
elsn |
|- ( ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } <-> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = .0. ) |
18 |
15 17
|
sylibr |
|- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } ) |
19 |
18
|
ralrimivva |
|- ( G e. Grp -> A. x e. ( Base ` G ) A. y e. { .0. } ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } ) |
20 |
6 7 12
|
isnsg3 |
|- ( { .0. } e. ( NrmSGrp ` G ) <-> ( { .0. } e. ( SubGrp ` G ) /\ A. x e. ( Base ` G ) A. y e. { .0. } ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } ) ) |
21 |
2 19 20
|
sylanbrc |
|- ( G e. Grp -> { .0. } e. ( NrmSGrp ` G ) ) |