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Metamath Proof Explorer

Theorem ngpds3r

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 ngpds3r 
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( .0. D ( B .- A ) ) )

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 ngpxms 
 |-  ( G e. NrmGrp -> G e. *MetSp )
6 1 4 xmssym 
 |-  ( ( G e. *MetSp /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
7 5 6 syl3an1 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
8 1 2 3 4 ngpds3 
 |-  ( ( G e. NrmGrp /\ B e. X /\ A e. X ) -> ( B D A ) = ( .0. D ( B .- A ) ) )
9 8 3com23 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( B D A ) = ( .0. D ( B .- A ) ) )
10 7 9 eqtrd 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( .0. D ( B .- A ) ) )