Step |
Hyp |
Ref |
Expression |
1 |
|
nv0id.1 |
|- X = ( BaseSet ` U ) |
2 |
|
nv0id.2 |
|- G = ( +v ` U ) |
3 |
|
nv0id.6 |
|- Z = ( 0vec ` U ) |
4 |
2 3
|
0vfval |
|- ( U e. NrmCVec -> Z = ( GId ` G ) ) |
5 |
4
|
oveq2d |
|- ( U e. NrmCVec -> ( A G Z ) = ( A G ( GId ` G ) ) ) |
6 |
5
|
adantr |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( A G Z ) = ( A G ( GId ` G ) ) ) |
7 |
2
|
nvgrp |
|- ( U e. NrmCVec -> G e. GrpOp ) |
8 |
1 2
|
bafval |
|- X = ran G |
9 |
|
eqid |
|- ( GId ` G ) = ( GId ` G ) |
10 |
8 9
|
grporid |
|- ( ( G e. GrpOp /\ A e. X ) -> ( A G ( GId ` G ) ) = A ) |
11 |
7 10
|
sylan |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( A G ( GId ` G ) ) = A ) |
12 |
6 11
|
eqtrd |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( A G Z ) = A ) |