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 ) ) )