Step |
Hyp |
Ref |
Expression |
1 |
|
grpidval.b |
|- B = ( Base ` G ) |
2 |
|
grpidval.p |
|- .+ = ( +g ` G ) |
3 |
|
grpidval.o |
|- .0. = ( 0g ` G ) |
4 |
|
fveq2 |
|- ( g = G -> ( Base ` g ) = ( Base ` G ) ) |
5 |
4 1
|
eqtr4di |
|- ( g = G -> ( Base ` g ) = B ) |
6 |
5
|
eleq2d |
|- ( g = G -> ( e e. ( Base ` g ) <-> e e. B ) ) |
7 |
|
fveq2 |
|- ( g = G -> ( +g ` g ) = ( +g ` G ) ) |
8 |
7 2
|
eqtr4di |
|- ( g = G -> ( +g ` g ) = .+ ) |
9 |
8
|
oveqd |
|- ( g = G -> ( e ( +g ` g ) x ) = ( e .+ x ) ) |
10 |
9
|
eqeq1d |
|- ( g = G -> ( ( e ( +g ` g ) x ) = x <-> ( e .+ x ) = x ) ) |
11 |
8
|
oveqd |
|- ( g = G -> ( x ( +g ` g ) e ) = ( x .+ e ) ) |
12 |
11
|
eqeq1d |
|- ( g = G -> ( ( x ( +g ` g ) e ) = x <-> ( x .+ e ) = x ) ) |
13 |
10 12
|
anbi12d |
|- ( g = G -> ( ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) <-> ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) |
14 |
5 13
|
raleqbidv |
|- ( g = G -> ( A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) <-> A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) |
15 |
6 14
|
anbi12d |
|- ( g = G -> ( ( e e. ( Base ` g ) /\ A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) ) <-> ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) ) |
16 |
15
|
iotabidv |
|- ( g = G -> ( iota e ( e e. ( Base ` g ) /\ A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) ) ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) ) |
17 |
|
df-0g |
|- 0g = ( g e. _V |-> ( iota e ( e e. ( Base ` g ) /\ A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) ) ) ) |
18 |
|
iotaex |
|- ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) e. _V |
19 |
16 17 18
|
fvmpt |
|- ( G e. _V -> ( 0g ` G ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) ) |
20 |
|
fvprc |
|- ( -. G e. _V -> ( 0g ` G ) = (/) ) |
21 |
|
euex |
|- ( E! e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> E. e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) |
22 |
|
n0i |
|- ( e e. B -> -. B = (/) ) |
23 |
|
fvprc |
|- ( -. G e. _V -> ( Base ` G ) = (/) ) |
24 |
1 23
|
eqtrid |
|- ( -. G e. _V -> B = (/) ) |
25 |
22 24
|
nsyl2 |
|- ( e e. B -> G e. _V ) |
26 |
25
|
adantr |
|- ( ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> G e. _V ) |
27 |
26
|
exlimiv |
|- ( E. e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> G e. _V ) |
28 |
21 27
|
syl |
|- ( E! e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> G e. _V ) |
29 |
|
iotanul |
|- ( -. E! e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) = (/) ) |
30 |
28 29
|
nsyl5 |
|- ( -. G e. _V -> ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) = (/) ) |
31 |
20 30
|
eqtr4d |
|- ( -. G e. _V -> ( 0g ` G ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) ) |
32 |
19 31
|
pm2.61i |
|- ( 0g ` G ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) |
33 |
3 32
|
eqtri |
|- .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) |