Step |
Hyp |
Ref |
Expression |
1 |
|
nrmtngdist.t |
|- T = ( G toNrmGrp ( norm ` G ) ) |
2 |
|
ngpgrp |
|- ( G e. NrmGrp -> G e. Grp ) |
3 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
4 |
1 3
|
nrmtngdist |
|- ( G e. NrmGrp -> ( dist ` T ) = ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) ) ) |
5 |
|
eqid |
|- ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) ) = ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) ) |
6 |
3 5
|
ngpmet |
|- ( G e. NrmGrp -> ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) ) e. ( Met ` ( Base ` G ) ) ) |
7 |
4 6
|
eqeltrd |
|- ( G e. NrmGrp -> ( dist ` T ) e. ( Met ` ( Base ` G ) ) ) |
8 |
|
eqid |
|- ( norm ` G ) = ( norm ` G ) |
9 |
3 8
|
nmf |
|- ( G e. NrmGrp -> ( norm ` G ) : ( Base ` G ) --> RR ) |
10 |
|
eqid |
|- ( dist ` T ) = ( dist ` T ) |
11 |
1 3 10
|
tngngp2 |
|- ( ( norm ` G ) : ( Base ` G ) --> RR -> ( T e. NrmGrp <-> ( G e. Grp /\ ( dist ` T ) e. ( Met ` ( Base ` G ) ) ) ) ) |
12 |
9 11
|
syl |
|- ( G e. NrmGrp -> ( T e. NrmGrp <-> ( G e. Grp /\ ( dist ` T ) e. ( Met ` ( Base ` G ) ) ) ) ) |
13 |
2 7 12
|
mpbir2and |
|- ( G e. NrmGrp -> T e. NrmGrp ) |
14 |
2 9
|
jca |
|- ( G e. NrmGrp -> ( G e. Grp /\ ( norm ` G ) : ( Base ` G ) --> RR ) ) |
15 |
|
reex |
|- RR e. _V |
16 |
1 3 15
|
tngnm |
|- ( ( G e. Grp /\ ( norm ` G ) : ( Base ` G ) --> RR ) -> ( norm ` G ) = ( norm ` T ) ) |
17 |
14 16
|
syl |
|- ( G e. NrmGrp -> ( norm ` G ) = ( norm ` T ) ) |
18 |
17
|
eqcomd |
|- ( G e. NrmGrp -> ( norm ` T ) = ( norm ` G ) ) |
19 |
13 18
|
jca |
|- ( G e. NrmGrp -> ( T e. NrmGrp /\ ( norm ` T ) = ( norm ` G ) ) ) |