Metamath Proof Explorer


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 = ( 0g𝐺 )
ngpds2.m = ( -g𝐺 )
ngpds2.d 𝐷 = ( dist ‘ 𝐺 )
Assertion ngpds2r ( ( 𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐵 𝐴 ) 𝐷 0 ) )

Proof

Step Hyp Ref Expression
1 ngpds2.x 𝑋 = ( Base ‘ 𝐺 )
2 ngpds2.z 0 = ( 0g𝐺 )
3 ngpds2.m = ( -g𝐺 )
4 ngpds2.d 𝐷 = ( dist ‘ 𝐺 )
5 ngpxms ( 𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp )
6 1 4 xmssym ( ( 𝐺 ∈ ∞MetSp ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐴 ) )
7 5 6 syl3an1 ( ( 𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐴 ) )
8 1 2 3 4 ngpds2 ( ( 𝐺 ∈ NrmGrp ∧ 𝐵𝑋𝐴𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( ( 𝐵 𝐴 ) 𝐷 0 ) )
9 8 3com23 ( ( 𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( ( 𝐵 𝐴 ) 𝐷 0 ) )
10 7 9 eqtrd ( ( 𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐵 𝐴 ) 𝐷 0 ) )