Metamath Proof Explorer

Theorem ngpds3

Description: Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses ngpds2.x 
|- X = ( Base ` G )
ngpds2.z 
|- .0. = ( 0g ` G )
ngpds2.m 
|- .- = ( -g ` G )
ngpds2.d 
|- D = ( dist ` G )
Assertion ngpds3 
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( .0. D ( A .- B ) ) )

Proof

Step Hyp Ref Expression
1 ngpds2.x 
 |-  X = ( Base ` G )
2 ngpds2.z 
 |-  .0. = ( 0g ` G )
3 ngpds2.m 
 |-  .- = ( -g ` G )
4 ngpds2.d 
 |-  D = ( dist ` G )
5 1 2 3 4 ngpds2 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( ( A .- B ) D .0. ) )
6 ngpxms 
 |-  ( G e. NrmGrp -> G e. *MetSp )
7 6 3ad2ant1 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> G e. *MetSp )
8 ngpgrp 
 |-  ( G e. NrmGrp -> G e. Grp )
9 1 3 grpsubcl 
 |-  ( ( G e. Grp /\ A e. X /\ B e. X ) -> ( A .- B ) e. X )
10 8 9 syl3an1 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A .- B ) e. X )
11 8 3ad2ant1 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> G e. Grp )
12 1 2 grpidcl 
 |-  ( G e. Grp -> .0. e. X )
13 11 12 syl 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> .0. e. X )
14 1 4 xmssym 
 |-  ( ( G e. *MetSp /\ ( A .- B ) e. X /\ .0. e. X ) -> ( ( A .- B ) D .0. ) = ( .0. D ( A .- B ) ) )
15 7 10 13 14 syl3anc 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( ( A .- B ) D .0. ) = ( .0. D ( A .- B ) ) )
16 5 15 eqtrd 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( .0. D ( A .- B ) ) )