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 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 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 ∈ 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	ngpds2	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( ( 𝐵 − 𝐴 ) 𝐷 0 ) )
9	8	3com23	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( ( 𝐵 − 𝐴 ) 𝐷 0 ) )
10	7 9	eqtrd	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐵 − 𝐴 ) 𝐷 0 ) )