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Metamath Proof Explorer

Theorem ngpdsr

Description: Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses ngpds.n 
|- N = ( norm ` G )
ngpds.x 
|- X = ( Base ` G )
ngpds.m 
|- .- = ( -g ` G )
ngpds.d 
|- D = ( dist ` G )
Assertion ngpdsr 
|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( B .- A ) ) )

Proof

Step Hyp Ref Expression
1 ngpds.n 
 |-  N = ( norm ` G )
2 ngpds.x 
 |-  X = ( Base ` G )
3 ngpds.m 
 |-  .- = ( -g ` G )
4 ngpds.d 
 |-  D = ( dist ` G )
5 ngpxms 
 |-  ( G e. NrmGrp -> G e. *MetSp )
6 2 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 ngpds 
 |-  ( ( G e. NrmGrp /\ B e. X /\ A e. X ) -> ( B D A ) = ( N ` ( B .- A ) ) )
9 8 3com23 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( B D A ) = ( N ` ( B .- A ) ) )
10 7 9 eqtrd 
 |-  ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( B .- A ) ) )