Theorem ngpds3

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	ngpds3	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 0 𝐷 ( 𝐴 − 𝐵 ) ) )

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	1 2 3 4	ngpds2	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) )
6		ngpxms	⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp )
7	6	3ad2ant1	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐺 ∈ ∞MetSp )
8		ngpgrp	⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ Grp )
9	1 3	grpsubcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 − 𝐵 ) ∈ 𝑋 )
10	8 9	syl3an1	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 − 𝐵 ) ∈ 𝑋 )
11	8	3ad2ant1	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐺 ∈ Grp )
12	1 2	grpidcl	⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝑋 )
13	11 12	syl	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 0 ∈ 𝑋 )
14	1 4	xmssym	⊢ ( ( 𝐺 ∈ ∞MetSp ∧ ( 𝐴 − 𝐵 ) ∈ 𝑋 ∧ 0 ∈ 𝑋 ) → ( ( 𝐴 − 𝐵 ) 𝐷 0 ) = ( 0 𝐷 ( 𝐴 − 𝐵 ) ) )
15	7 10 13 14	syl3anc	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 − 𝐵 ) 𝐷 0 ) = ( 0 𝐷 ( 𝐴 − 𝐵 ) ) )
16	5 15	eqtrd	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 0 𝐷 ( 𝐴 − 𝐵 ) ) )