Step |
Hyp |
Ref |
Expression |
1 |
|
idnghm.2 |
|- V = ( Base ` S ) |
2 |
|
eqid |
|- ( S normOp S ) = ( S normOp S ) |
3 |
|
eqid |
|- ( 0g ` S ) = ( 0g ` S ) |
4 |
2 1 3
|
nmoid |
|- ( ( S e. NrmGrp /\ { ( 0g ` S ) } C. V ) -> ( ( S normOp S ) ` ( _I |` V ) ) = 1 ) |
5 |
|
1re |
|- 1 e. RR |
6 |
4 5
|
eqeltrdi |
|- ( ( S e. NrmGrp /\ { ( 0g ` S ) } C. V ) -> ( ( S normOp S ) ` ( _I |` V ) ) e. RR ) |
7 |
|
eleq2 |
|- ( { ( 0g ` S ) } = V -> ( x e. { ( 0g ` S ) } <-> x e. V ) ) |
8 |
7
|
biimpar |
|- ( ( { ( 0g ` S ) } = V /\ x e. V ) -> x e. { ( 0g ` S ) } ) |
9 |
|
elsni |
|- ( x e. { ( 0g ` S ) } -> x = ( 0g ` S ) ) |
10 |
8 9
|
syl |
|- ( ( { ( 0g ` S ) } = V /\ x e. V ) -> x = ( 0g ` S ) ) |
11 |
10
|
mpteq2dva |
|- ( { ( 0g ` S ) } = V -> ( x e. V |-> x ) = ( x e. V |-> ( 0g ` S ) ) ) |
12 |
|
mptresid |
|- ( _I |` V ) = ( x e. V |-> x ) |
13 |
|
fconstmpt |
|- ( V X. { ( 0g ` S ) } ) = ( x e. V |-> ( 0g ` S ) ) |
14 |
11 12 13
|
3eqtr4g |
|- ( { ( 0g ` S ) } = V -> ( _I |` V ) = ( V X. { ( 0g ` S ) } ) ) |
15 |
14
|
fveq2d |
|- ( { ( 0g ` S ) } = V -> ( ( S normOp S ) ` ( _I |` V ) ) = ( ( S normOp S ) ` ( V X. { ( 0g ` S ) } ) ) ) |
16 |
2 1 3
|
nmo0 |
|- ( ( S e. NrmGrp /\ S e. NrmGrp ) -> ( ( S normOp S ) ` ( V X. { ( 0g ` S ) } ) ) = 0 ) |
17 |
16
|
anidms |
|- ( S e. NrmGrp -> ( ( S normOp S ) ` ( V X. { ( 0g ` S ) } ) ) = 0 ) |
18 |
15 17
|
sylan9eqr |
|- ( ( S e. NrmGrp /\ { ( 0g ` S ) } = V ) -> ( ( S normOp S ) ` ( _I |` V ) ) = 0 ) |
19 |
|
0re |
|- 0 e. RR |
20 |
18 19
|
eqeltrdi |
|- ( ( S e. NrmGrp /\ { ( 0g ` S ) } = V ) -> ( ( S normOp S ) ` ( _I |` V ) ) e. RR ) |
21 |
|
ngpgrp |
|- ( S e. NrmGrp -> S e. Grp ) |
22 |
1 3
|
grpidcl |
|- ( S e. Grp -> ( 0g ` S ) e. V ) |
23 |
21 22
|
syl |
|- ( S e. NrmGrp -> ( 0g ` S ) e. V ) |
24 |
23
|
snssd |
|- ( S e. NrmGrp -> { ( 0g ` S ) } C_ V ) |
25 |
|
sspss |
|- ( { ( 0g ` S ) } C_ V <-> ( { ( 0g ` S ) } C. V \/ { ( 0g ` S ) } = V ) ) |
26 |
24 25
|
sylib |
|- ( S e. NrmGrp -> ( { ( 0g ` S ) } C. V \/ { ( 0g ` S ) } = V ) ) |
27 |
6 20 26
|
mpjaodan |
|- ( S e. NrmGrp -> ( ( S normOp S ) ` ( _I |` V ) ) e. RR ) |
28 |
|
id |
|- ( S e. NrmGrp -> S e. NrmGrp ) |
29 |
1
|
idghm |
|- ( S e. Grp -> ( _I |` V ) e. ( S GrpHom S ) ) |
30 |
21 29
|
syl |
|- ( S e. NrmGrp -> ( _I |` V ) e. ( S GrpHom S ) ) |
31 |
2
|
isnghm2 |
|- ( ( S e. NrmGrp /\ S e. NrmGrp /\ ( _I |` V ) e. ( S GrpHom S ) ) -> ( ( _I |` V ) e. ( S NGHom S ) <-> ( ( S normOp S ) ` ( _I |` V ) ) e. RR ) ) |
32 |
28 30 31
|
mpd3an23 |
|- ( S e. NrmGrp -> ( ( _I |` V ) e. ( S NGHom S ) <-> ( ( S normOp S ) ` ( _I |` V ) ) e. RR ) ) |
33 |
27 32
|
mpbird |
|- ( S e. NrmGrp -> ( _I |` V ) e. ( S NGHom S ) ) |