Step |
Hyp |
Ref |
Expression |
1 |
|
nmf.x |
|- X = ( Base ` G ) |
2 |
|
nmf.n |
|- N = ( norm ` G ) |
3 |
|
nmmtri.m |
|- .- = ( -g ` G ) |
4 |
|
eqid |
|- ( dist ` G ) = ( dist ` G ) |
5 |
2 1 3 4
|
ngpds |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A ( dist ` G ) B ) = ( N ` ( A .- B ) ) ) |
6 |
|
ngpms |
|- ( G e. NrmGrp -> G e. MetSp ) |
7 |
6
|
3ad2ant1 |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> G e. MetSp ) |
8 |
|
simp2 |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> A e. X ) |
9 |
|
simp3 |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> B e. X ) |
10 |
|
ngpgrp |
|- ( G e. NrmGrp -> G e. Grp ) |
11 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
12 |
1 11
|
grpidcl |
|- ( G e. Grp -> ( 0g ` G ) e. X ) |
13 |
10 12
|
syl |
|- ( G e. NrmGrp -> ( 0g ` G ) e. X ) |
14 |
13
|
3ad2ant1 |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( 0g ` G ) e. X ) |
15 |
1 4
|
mstri3 |
|- ( ( G e. MetSp /\ ( A e. X /\ B e. X /\ ( 0g ` G ) e. X ) ) -> ( A ( dist ` G ) B ) <_ ( ( A ( dist ` G ) ( 0g ` G ) ) + ( B ( dist ` G ) ( 0g ` G ) ) ) ) |
16 |
7 8 9 14 15
|
syl13anc |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A ( dist ` G ) B ) <_ ( ( A ( dist ` G ) ( 0g ` G ) ) + ( B ( dist ` G ) ( 0g ` G ) ) ) ) |
17 |
2 1 11 4
|
nmval |
|- ( A e. X -> ( N ` A ) = ( A ( dist ` G ) ( 0g ` G ) ) ) |
18 |
17
|
3ad2ant2 |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( N ` A ) = ( A ( dist ` G ) ( 0g ` G ) ) ) |
19 |
2 1 11 4
|
nmval |
|- ( B e. X -> ( N ` B ) = ( B ( dist ` G ) ( 0g ` G ) ) ) |
20 |
19
|
3ad2ant3 |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( N ` B ) = ( B ( dist ` G ) ( 0g ` G ) ) ) |
21 |
18 20
|
oveq12d |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( ( N ` A ) + ( N ` B ) ) = ( ( A ( dist ` G ) ( 0g ` G ) ) + ( B ( dist ` G ) ( 0g ` G ) ) ) ) |
22 |
16 21
|
breqtrrd |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A ( dist ` G ) B ) <_ ( ( N ` A ) + ( N ` B ) ) ) |
23 |
5 22
|
eqbrtrrd |
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( N ` ( A .- B ) ) <_ ( ( N ` A ) + ( N ` B ) ) ) |