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Metamath Proof Explorer

Theorem ngpds2

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 ngpds2 
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( ( A .- B ) D .0. ) )

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 eqid 
 |-  ( norm ` G ) = ( norm ` G )
6 5 1 3 4 ngpds 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( ( norm ` G ) ` ( A .- B ) ) )
7 ngpgrp 
 |-  ( G e. NrmGrp -> G e. Grp )
8 1 3 grpsubcl 
 |-  ( ( G e. Grp /\ A e. X /\ B e. X ) -> ( A .- B ) e. X )
9 7 8 syl3an1 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A .- B ) e. X )
10 5 1 2 4 nmval 
 |-  ( ( A .- B ) e. X -> ( ( norm ` G ) ` ( A .- B ) ) = ( ( A .- B ) D .0. ) )
11 9 10 syl 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( ( norm ` G ) ` ( A .- B ) ) = ( ( A .- B ) D .0. ) )
12 6 11 eqtrd 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( ( A .- B ) D .0. ) )