Theorem ttgds

Description: The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019)

	Ref	Expression
Hypotheses	ttgval.n	⊢ 𝐺 = ( toTG ‘ 𝐻 )
	ttgds.1	⊢ 𝐷 = ( dist ‘ 𝐻 )
Assertion	ttgds	⊢ 𝐷 = ( dist ‘ 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		ttgval.n	⊢ 𝐺 = ( toTG ‘ 𝐻 )
2		ttgds.1	⊢ 𝐷 = ( dist ‘ 𝐻 )
3		df-ds	⊢ dist = Slot ; 1 2
4		1nn0	⊢ 1 ∈ ℕ0
5		2nn	⊢ 2 ∈ ℕ
6	4 5	decnncl	⊢ ; 1 2 ∈ ℕ
7		2nn0	⊢ 2 ∈ ℕ0
8		6nn	⊢ 6 ∈ ℕ
9		2lt6	⊢ 2 < 6
10	4 7 8 9	declt	⊢ ; 1 2 < ; 1 6
11	1 3 6 10	ttglem	⊢ ( dist ‘ 𝐻 ) = ( dist ‘ 𝐺 )
12	2 11	eqtri	⊢ 𝐷 = ( dist ‘ 𝐺 )