Step |
Hyp |
Ref |
Expression |
1 |
|
qusgrp.h |
|- H = ( G /s ( G ~QG S ) ) |
2 |
1
|
a1i |
|- ( S e. ( NrmSGrp ` G ) -> H = ( G /s ( G ~QG S ) ) ) |
3 |
|
eqidd |
|- ( S e. ( NrmSGrp ` G ) -> ( Base ` G ) = ( Base ` G ) ) |
4 |
|
eqidd |
|- ( S e. ( NrmSGrp ` G ) -> ( +g ` G ) = ( +g ` G ) ) |
5 |
|
nsgsubg |
|- ( S e. ( NrmSGrp ` G ) -> S e. ( SubGrp ` G ) ) |
6 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
7 |
|
eqid |
|- ( G ~QG S ) = ( G ~QG S ) |
8 |
6 7
|
eqger |
|- ( S e. ( SubGrp ` G ) -> ( G ~QG S ) Er ( Base ` G ) ) |
9 |
5 8
|
syl |
|- ( S e. ( NrmSGrp ` G ) -> ( G ~QG S ) Er ( Base ` G ) ) |
10 |
|
subgrcl |
|- ( S e. ( SubGrp ` G ) -> G e. Grp ) |
11 |
5 10
|
syl |
|- ( S e. ( NrmSGrp ` G ) -> G e. Grp ) |
12 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
13 |
6 7 12
|
eqgcpbl |
|- ( S e. ( NrmSGrp ` G ) -> ( ( a ( G ~QG S ) c /\ b ( G ~QG S ) d ) -> ( a ( +g ` G ) b ) ( G ~QG S ) ( c ( +g ` G ) d ) ) ) |
14 |
6 12
|
grpcl |
|- ( ( G e. Grp /\ u e. ( Base ` G ) /\ v e. ( Base ` G ) ) -> ( u ( +g ` G ) v ) e. ( Base ` G ) ) |
15 |
11 14
|
syl3an1 |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) /\ v e. ( Base ` G ) ) -> ( u ( +g ` G ) v ) e. ( Base ` G ) ) |
16 |
9
|
adantr |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> ( G ~QG S ) Er ( Base ` G ) ) |
17 |
11
|
adantr |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> G e. Grp ) |
18 |
|
simpr1 |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> u e. ( Base ` G ) ) |
19 |
|
simpr2 |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> v e. ( Base ` G ) ) |
20 |
17 18 19 14
|
syl3anc |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> ( u ( +g ` G ) v ) e. ( Base ` G ) ) |
21 |
|
simpr3 |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> w e. ( Base ` G ) ) |
22 |
6 12
|
grpcl |
|- ( ( G e. Grp /\ ( u ( +g ` G ) v ) e. ( Base ` G ) /\ w e. ( Base ` G ) ) -> ( ( u ( +g ` G ) v ) ( +g ` G ) w ) e. ( Base ` G ) ) |
23 |
17 20 21 22
|
syl3anc |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> ( ( u ( +g ` G ) v ) ( +g ` G ) w ) e. ( Base ` G ) ) |
24 |
16 23
|
erref |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> ( ( u ( +g ` G ) v ) ( +g ` G ) w ) ( G ~QG S ) ( ( u ( +g ` G ) v ) ( +g ` G ) w ) ) |
25 |
6 12
|
grpass |
|- ( ( G e. Grp /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> ( ( u ( +g ` G ) v ) ( +g ` G ) w ) = ( u ( +g ` G ) ( v ( +g ` G ) w ) ) ) |
26 |
11 25
|
sylan |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> ( ( u ( +g ` G ) v ) ( +g ` G ) w ) = ( u ( +g ` G ) ( v ( +g ` G ) w ) ) ) |
27 |
24 26
|
breqtrd |
|- ( ( S e. ( NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> ( ( u ( +g ` G ) v ) ( +g ` G ) w ) ( G ~QG S ) ( u ( +g ` G ) ( v ( +g ` G ) w ) ) ) |
28 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
29 |
6 28
|
grpidcl |
|- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) ) |
30 |
11 29
|
syl |
|- ( S e. ( NrmSGrp ` G ) -> ( 0g ` G ) e. ( Base ` G ) ) |
31 |
6 12 28
|
grplid |
|- ( ( G e. Grp /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) = u ) |
32 |
11 31
|
sylan |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) = u ) |
33 |
9
|
adantr |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( G ~QG S ) Er ( Base ` G ) ) |
34 |
|
simpr |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> u e. ( Base ` G ) ) |
35 |
33 34
|
erref |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> u ( G ~QG S ) u ) |
36 |
32 35
|
eqbrtrd |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) ( G ~QG S ) u ) |
37 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
38 |
6 37
|
grpinvcl |
|- ( ( G e. Grp /\ u e. ( Base ` G ) ) -> ( ( invg ` G ) ` u ) e. ( Base ` G ) ) |
39 |
11 38
|
sylan |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( invg ` G ) ` u ) e. ( Base ` G ) ) |
40 |
6 12 28 37
|
grplinv |
|- ( ( G e. Grp /\ u e. ( Base ` G ) ) -> ( ( ( invg ` G ) ` u ) ( +g ` G ) u ) = ( 0g ` G ) ) |
41 |
11 40
|
sylan |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( ( invg ` G ) ` u ) ( +g ` G ) u ) = ( 0g ` G ) ) |
42 |
30
|
adantr |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( 0g ` G ) e. ( Base ` G ) ) |
43 |
33 42
|
erref |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( 0g ` G ) ( G ~QG S ) ( 0g ` G ) ) |
44 |
41 43
|
eqbrtrd |
|- ( ( S e. ( NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( ( invg ` G ) ` u ) ( +g ` G ) u ) ( G ~QG S ) ( 0g ` G ) ) |
45 |
2 3 4 9 11 13 15 27 30 36 39 44
|
qusgrp2 |
|- ( S e. ( NrmSGrp ` G ) -> ( H e. Grp /\ [ ( 0g ` G ) ] ( G ~QG S ) = ( 0g ` H ) ) ) |
46 |
45
|
simpld |
|- ( S e. ( NrmSGrp ` G ) -> H e. Grp ) |