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. ) )