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 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 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 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( norm ‘ 𝐺 ) ‘ ( 𝐴 − 𝐵 ) ) )
7		ngpgrp	⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ Grp )
8	1 3	grpsubcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 − 𝐵 ) ∈ 𝑋 )
9	7 8	syl3an1	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 − 𝐵 ) ∈ 𝑋 )
10	5 1 2 4	nmval	⊢ ( ( 𝐴 − 𝐵 ) ∈ 𝑋 → ( ( norm ‘ 𝐺 ) ‘ ( 𝐴 − 𝐵 ) ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) )
11	9 10	syl	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( norm ‘ 𝐺 ) ‘ ( 𝐴 − 𝐵 ) ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) )
12	6 11	eqtrd	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) )