Step |
Hyp |
Ref |
Expression |
1 |
|
oppgbas.1 |
|- O = ( oppG ` R ) |
2 |
|
oppginv.2 |
|- I = ( invg ` R ) |
3 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
4 |
3 2
|
grpinvf |
|- ( R e. Grp -> I : ( Base ` R ) --> ( Base ` R ) ) |
5 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
6 |
|
eqid |
|- ( +g ` O ) = ( +g ` O ) |
7 |
5 1 6
|
oppgplus |
|- ( ( I ` x ) ( +g ` O ) x ) = ( x ( +g ` R ) ( I ` x ) ) |
8 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
9 |
3 5 8 2
|
grprinv |
|- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( x ( +g ` R ) ( I ` x ) ) = ( 0g ` R ) ) |
10 |
7 9
|
syl5eq |
|- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( ( I ` x ) ( +g ` O ) x ) = ( 0g ` R ) ) |
11 |
10
|
ralrimiva |
|- ( R e. Grp -> A. x e. ( Base ` R ) ( ( I ` x ) ( +g ` O ) x ) = ( 0g ` R ) ) |
12 |
1
|
oppggrp |
|- ( R e. Grp -> O e. Grp ) |
13 |
1 3
|
oppgbas |
|- ( Base ` R ) = ( Base ` O ) |
14 |
1 8
|
oppgid |
|- ( 0g ` R ) = ( 0g ` O ) |
15 |
|
eqid |
|- ( invg ` O ) = ( invg ` O ) |
16 |
13 6 14 15
|
isgrpinv |
|- ( O e. Grp -> ( ( I : ( Base ` R ) --> ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( I ` x ) ( +g ` O ) x ) = ( 0g ` R ) ) <-> ( invg ` O ) = I ) ) |
17 |
12 16
|
syl |
|- ( R e. Grp -> ( ( I : ( Base ` R ) --> ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( I ` x ) ( +g ` O ) x ) = ( 0g ` R ) ) <-> ( invg ` O ) = I ) ) |
18 |
4 11 17
|
mpbi2and |
|- ( R e. Grp -> ( invg ` O ) = I ) |
19 |
18
|
eqcomd |
|- ( R e. Grp -> I = ( invg ` O ) ) |