Theorem trkgitv

Description: The congruence relation in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017)

	Ref	Expression
Hypothesis	trkgstr.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},U⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩,⟨\mathrm{Itv}\left(\mathrm{ndx}\right),I⟩\right\}$$
Assertion	trkgitv	$$⊢I\in V\to I=\mathrm{Itv}\left(W\right)$$

Proof

Step	Hyp	Ref	Expression
1		trkgstr.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},U⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩,⟨\mathrm{Itv}\left(\mathrm{ndx}\right),I⟩\right\}$$
2	1	trkgstr	$$⊢W\mathrm{Struct}⟨1,16⟩$$
3		itvid	$$⊢\mathrm{Itv}=\mathrm{Slot}\mathrm{Itv}\left(\mathrm{ndx}\right)$$
4		snsstp3	$$⊢\left\{⟨\mathrm{Itv}\left(\mathrm{ndx}\right),I⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},U⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩,⟨\mathrm{Itv}\left(\mathrm{ndx}\right),I⟩\right\}$$
5	4 1	sseqtrri	$$⊢\left\{⟨\mathrm{Itv}\left(\mathrm{ndx}\right),I⟩\right\}\subseteq W$$
6	2 3 5	strfv	$$⊢I\in V\to I=\mathrm{Itv}\left(W\right)$$