Step |
Hyp |
Ref |
Expression |
1 |
|
grpidssd.m |
|- ( ph -> M e. Grp ) |
2 |
|
grpidssd.s |
|- ( ph -> S e. Grp ) |
3 |
|
grpidssd.b |
|- B = ( Base ` S ) |
4 |
|
grpidssd.c |
|- ( ph -> B C_ ( Base ` M ) ) |
5 |
|
grpidssd.o |
|- ( ph -> A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) ) |
6 |
|
eqid |
|- ( 0g ` S ) = ( 0g ` S ) |
7 |
3 6
|
grpidcl |
|- ( S e. Grp -> ( 0g ` S ) e. B ) |
8 |
2 7
|
syl |
|- ( ph -> ( 0g ` S ) e. B ) |
9 |
|
oveq1 |
|- ( x = ( 0g ` S ) -> ( x ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) y ) ) |
10 |
|
oveq1 |
|- ( x = ( 0g ` S ) -> ( x ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) ) |
11 |
9 10
|
eqeq12d |
|- ( x = ( 0g ` S ) -> ( ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) <-> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) ) ) |
12 |
|
oveq2 |
|- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) ) |
13 |
|
oveq2 |
|- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) ) |
14 |
12 13
|
eqeq12d |
|- ( y = ( 0g ` S ) -> ( ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) <-> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) ) ) |
15 |
11 14
|
rspc2va |
|- ( ( ( ( 0g ` S ) e. B /\ ( 0g ` S ) e. B ) /\ A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) ) -> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) ) |
16 |
8 8 5 15
|
syl21anc |
|- ( ph -> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) ) |
17 |
|
eqid |
|- ( +g ` S ) = ( +g ` S ) |
18 |
3 17 6
|
grplid |
|- ( ( S e. Grp /\ ( 0g ` S ) e. B ) -> ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) = ( 0g ` S ) ) |
19 |
2 7 18
|
syl2anc2 |
|- ( ph -> ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) = ( 0g ` S ) ) |
20 |
16 19
|
eqtrd |
|- ( ph -> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( 0g ` S ) ) |
21 |
4 8
|
sseldd |
|- ( ph -> ( 0g ` S ) e. ( Base ` M ) ) |
22 |
|
eqid |
|- ( Base ` M ) = ( Base ` M ) |
23 |
|
eqid |
|- ( +g ` M ) = ( +g ` M ) |
24 |
|
eqid |
|- ( 0g ` M ) = ( 0g ` M ) |
25 |
22 23 24
|
grpidlcan |
|- ( ( M e. Grp /\ ( 0g ` S ) e. ( Base ` M ) /\ ( 0g ` S ) e. ( Base ` M ) ) -> ( ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( 0g ` S ) <-> ( 0g ` S ) = ( 0g ` M ) ) ) |
26 |
1 21 21 25
|
syl3anc |
|- ( ph -> ( ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( 0g ` S ) <-> ( 0g ` S ) = ( 0g ` M ) ) ) |
27 |
20 26
|
mpbid |
|- ( ph -> ( 0g ` S ) = ( 0g ` M ) ) |
28 |
27
|
eqcomd |
|- ( ph -> ( 0g ` M ) = ( 0g ` S ) ) |