Theorem ngpds2r

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	ngpds2r	|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( ( B .- A ) 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		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	ngpds2	|- ( ( G e. NrmGrp /\ B e. X /\ A e. X ) -> ( B D A ) = ( ( B .- A ) D .0. ) )
9	8	3com23	|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( B D A ) = ( ( B .- A ) D .0. ) )
10	7 9	eqtrd	|- ( ( G e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( ( B .- A ) D .0. ) )