Step |
Hyp |
Ref |
Expression |
1 |
|
subginvcl.i |
|- I = ( invg ` G ) |
2 |
|
eqid |
|- ( G |`s S ) = ( G |`s S ) |
3 |
2
|
subggrp |
|- ( S e. ( SubGrp ` G ) -> ( G |`s S ) e. Grp ) |
4 |
|
simpr |
|- ( ( S e. ( SubGrp ` G ) /\ X e. S ) -> X e. S ) |
5 |
2
|
subgbas |
|- ( S e. ( SubGrp ` G ) -> S = ( Base ` ( G |`s S ) ) ) |
6 |
5
|
adantr |
|- ( ( S e. ( SubGrp ` G ) /\ X e. S ) -> S = ( Base ` ( G |`s S ) ) ) |
7 |
4 6
|
eleqtrd |
|- ( ( S e. ( SubGrp ` G ) /\ X e. S ) -> X e. ( Base ` ( G |`s S ) ) ) |
8 |
|
eqid |
|- ( Base ` ( G |`s S ) ) = ( Base ` ( G |`s S ) ) |
9 |
|
eqid |
|- ( invg ` ( G |`s S ) ) = ( invg ` ( G |`s S ) ) |
10 |
8 9
|
grpinvcl |
|- ( ( ( G |`s S ) e. Grp /\ X e. ( Base ` ( G |`s S ) ) ) -> ( ( invg ` ( G |`s S ) ) ` X ) e. ( Base ` ( G |`s S ) ) ) |
11 |
3 7 10
|
syl2an2r |
|- ( ( S e. ( SubGrp ` G ) /\ X e. S ) -> ( ( invg ` ( G |`s S ) ) ` X ) e. ( Base ` ( G |`s S ) ) ) |
12 |
2 1 9
|
subginv |
|- ( ( S e. ( SubGrp ` G ) /\ X e. S ) -> ( I ` X ) = ( ( invg ` ( G |`s S ) ) ` X ) ) |
13 |
11 12 6
|
3eltr4d |
|- ( ( S e. ( SubGrp ` G ) /\ X e. S ) -> ( I ` X ) e. S ) |