Step |
Hyp |
Ref |
Expression |
1 |
|
grpidinv.b |
|- B = ( Base ` G ) |
2 |
|
grpidinv.p |
|- .+ = ( +g ` G ) |
3 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
4 |
1 3
|
grpidcl |
|- ( G e. Grp -> ( 0g ` G ) e. B ) |
5 |
|
oveq1 |
|- ( u = ( 0g ` G ) -> ( u .+ x ) = ( ( 0g ` G ) .+ x ) ) |
6 |
5
|
eqeq1d |
|- ( u = ( 0g ` G ) -> ( ( u .+ x ) = x <-> ( ( 0g ` G ) .+ x ) = x ) ) |
7 |
|
oveq2 |
|- ( u = ( 0g ` G ) -> ( x .+ u ) = ( x .+ ( 0g ` G ) ) ) |
8 |
7
|
eqeq1d |
|- ( u = ( 0g ` G ) -> ( ( x .+ u ) = x <-> ( x .+ ( 0g ` G ) ) = x ) ) |
9 |
6 8
|
anbi12d |
|- ( u = ( 0g ` G ) -> ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) ) ) |
10 |
|
eqeq2 |
|- ( u = ( 0g ` G ) -> ( ( y .+ x ) = u <-> ( y .+ x ) = ( 0g ` G ) ) ) |
11 |
|
eqeq2 |
|- ( u = ( 0g ` G ) -> ( ( x .+ y ) = u <-> ( x .+ y ) = ( 0g ` G ) ) ) |
12 |
10 11
|
anbi12d |
|- ( u = ( 0g ` G ) -> ( ( ( y .+ x ) = u /\ ( x .+ y ) = u ) <-> ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) |
13 |
12
|
rexbidv |
|- ( u = ( 0g ` G ) -> ( E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) <-> E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) |
14 |
9 13
|
anbi12d |
|- ( u = ( 0g ` G ) -> ( ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) <-> ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) ) |
15 |
14
|
ralbidv |
|- ( u = ( 0g ` G ) -> ( A. x e. B ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) <-> A. x e. B ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) ) |
16 |
15
|
adantl |
|- ( ( G e. Grp /\ u = ( 0g ` G ) ) -> ( A. x e. B ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) <-> A. x e. B ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) ) |
17 |
1 2 3
|
grpidinv2 |
|- ( ( G e. Grp /\ x e. B ) -> ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) |
18 |
17
|
ralrimiva |
|- ( G e. Grp -> A. x e. B ( ( ( ( 0g ` G ) .+ x ) = x /\ ( x .+ ( 0g ` G ) ) = x ) /\ E. y e. B ( ( y .+ x ) = ( 0g ` G ) /\ ( x .+ y ) = ( 0g ` G ) ) ) ) |
19 |
4 16 18
|
rspcedvd |
|- ( G e. Grp -> E. u e. B A. x e. B ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) ) |