Theorem ngpdsr

Description: Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	ngpds.n	⊢ 𝑁 = ( norm ‘ 𝐺 )
	ngpds.x	⊢ 𝑋 = ( Base ‘ 𝐺 )
	ngpds.m	⊢ − = ( -g ‘ 𝐺 )
	ngpds.d	⊢ 𝐷 = ( dist ‘ 𝐺 )
Assertion	ngpdsr	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐵 − 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ngpds.n	⊢ 𝑁 = ( norm ‘ 𝐺 )
2		ngpds.x	⊢ 𝑋 = ( Base ‘ 𝐺 )
3		ngpds.m	⊢ − = ( -g ‘ 𝐺 )
4		ngpds.d	⊢ 𝐷 = ( dist ‘ 𝐺 )
5		ngpxms	⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp )
6	2 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	ngpds	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( 𝑁 ‘ ( 𝐵 − 𝐴 ) ) )
9	8	3com23	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( 𝑁 ‘ ( 𝐵 − 𝐴 ) ) )
10	7 9	eqtrd	⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐵 − 𝐴 ) ) )