Step |
Hyp |
Ref |
Expression |
1 |
|
nvmval.1 |
|- X = ( BaseSet ` U ) |
2 |
|
nvmval.2 |
|- G = ( +v ` U ) |
3 |
|
nvmval.4 |
|- S = ( .sOLD ` U ) |
4 |
|
nvmval.3 |
|- M = ( -v ` U ) |
5 |
2
|
nvgrp |
|- ( U e. NrmCVec -> G e. GrpOp ) |
6 |
1 2
|
bafval |
|- X = ran G |
7 |
|
eqid |
|- ( inv ` G ) = ( inv ` G ) |
8 |
|
eqid |
|- ( /g ` G ) = ( /g ` G ) |
9 |
6 7 8
|
grpodivval |
|- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A ( /g ` G ) B ) = ( A G ( ( inv ` G ) ` B ) ) ) |
10 |
5 9
|
syl3an1 |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A ( /g ` G ) B ) = ( A G ( ( inv ` G ) ` B ) ) ) |
11 |
1 2 4 8
|
nvm |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( A ( /g ` G ) B ) ) |
12 |
1 2 3 7
|
nvinv |
|- ( ( U e. NrmCVec /\ B e. X ) -> ( -u 1 S B ) = ( ( inv ` G ) ` B ) ) |
13 |
12
|
3adant2 |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S B ) = ( ( inv ` G ) ` B ) ) |
14 |
13
|
oveq2d |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A G ( -u 1 S B ) ) = ( A G ( ( inv ` G ) ` B ) ) ) |
15 |
10 11 14
|
3eqtr4d |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( A G ( -u 1 S B ) ) ) |