Step |
Hyp |
Ref |
Expression |
1 |
|
0ex |
|- (/) e. _V |
2 |
1
|
jctr |
|- ( G e. Grp -> ( G e. Grp /\ (/) e. _V ) ) |
3 |
|
f0 |
|- (/) : (/) --> (/) |
4 |
|
xp0 |
|- ( ( Base ` G ) X. (/) ) = (/) |
5 |
4
|
feq2i |
|- ( (/) : ( ( Base ` G ) X. (/) ) --> (/) <-> (/) : (/) --> (/) ) |
6 |
3 5
|
mpbir |
|- (/) : ( ( Base ` G ) X. (/) ) --> (/) |
7 |
|
ral0 |
|- A. x e. (/) ( ( ( 0g ` G ) (/) x ) = x /\ A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) (/) x ) = ( y (/) ( z (/) x ) ) ) |
8 |
6 7
|
pm3.2i |
|- ( (/) : ( ( Base ` G ) X. (/) ) --> (/) /\ A. x e. (/) ( ( ( 0g ` G ) (/) x ) = x /\ A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) (/) x ) = ( y (/) ( z (/) x ) ) ) ) |
9 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
10 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
11 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
12 |
9 10 11
|
isga |
|- ( (/) e. ( G GrpAct (/) ) <-> ( ( G e. Grp /\ (/) e. _V ) /\ ( (/) : ( ( Base ` G ) X. (/) ) --> (/) /\ A. x e. (/) ( ( ( 0g ` G ) (/) x ) = x /\ A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) (/) x ) = ( y (/) ( z (/) x ) ) ) ) ) ) |
13 |
2 8 12
|
sylanblrc |
|- ( G e. Grp -> (/) e. ( G GrpAct (/) ) ) |