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	$$⊢G={to𝒢}_{\mathrm{Tarski}}\left(H\right)$$
	ttgds.1	$$⊢D=\mathrm{dist}\left(H\right)$$
Assertion	ttgds	$$⊢D=\mathrm{dist}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		ttgval.n	$$⊢G={to𝒢}_{\mathrm{Tarski}}\left(H\right)$$
2		ttgds.1	$$⊢D=\mathrm{dist}\left(H\right)$$
3		df-ds	$$⊢\mathrm{dist}=\mathrm{Slot}12$$
4		1nn0	$$⊢1\in {\mathbb{N}}_0$$
5		2nn	$$⊢2\in \mathbb{N}$$
6	4 5	decnncl	$$⊢12\in \mathbb{N}$$
7		2nn0	$$⊢2\in {\mathbb{N}}_0$$
8		6nn	$$⊢6\in \mathbb{N}$$
9		2lt6	$$⊢2<6$$
10	4 7 8 9	declt	$$⊢12<16$$
11	1 3 6 10	ttglem	$$⊢\mathrm{dist}\left(H\right)=\mathrm{dist}\left(G\right)$$
12	2 11	eqtri	$$⊢D=\mathrm{dist}\left(G\right)$$