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 ˙ = 0 G
ngpds2.m - ˙ = - G
ngpds2.d D = dist G
Assertion ngpds3r G NrmGrp A X B X A D B = 0 ˙ D B - ˙ A

Proof

Step Hyp Ref Expression
1 ngpds2.x X = Base G
2 ngpds2.z 0 ˙ = 0 G
3 ngpds2.m - ˙ = - G
4 ngpds2.d D = dist G
5 ngpxms G NrmGrp G ∞MetSp
6 1 4 xmssym G ∞MetSp A X B X A D B = B D A
7 5 6 syl3an1 G NrmGrp A X B X A D B = B D A
8 1 2 3 4 ngpds3 G NrmGrp B X A X B D A = 0 ˙ D B - ˙ A
9 8 3com23 G NrmGrp A X B X B D A = 0 ˙ D B - ˙ A
10 7 9 eqtrd G NrmGrp A X B X A D B = 0 ˙ D B - ˙ A