Theorem legbtwn

Description: Deduce betweenness from "less than" relation. Corresponds loosely to Proposition 6.13 of Schwabhauser p. 45. (Contributed by Thierry Arnoux, 25-Aug-2019)

	Ref	Expression
Hypotheses	legval.p	$$⊢P={\mathrm{Base}}_G$$
	legval.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	legval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	legval.l	$$⊢\underset{\dot{}}{\le }={\le }_{𝒢}\left(G\right)$$
	legval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	legid.a	$$⊢\phi \to A\in P$$
	legid.b	$$⊢\phi \to B\in P$$
	legtrd.c	$$⊢\phi \to C\in P$$
	legtrd.d	$$⊢\phi \to D\in P$$
	legbtwn.1	$$⊢\phi \to \left(A\in \left(CIB\right)\vee B\in \left(CIA\right)\right)$$
	legbtwn.2	$$⊢\phi \to \left(C\underset{\dot{}}{-}A\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}B\right)$$
Assertion	legbtwn	$$⊢\phi \to A\in \left(CIB\right)$$

Proof

Step	Hyp	Ref	Expression
1		legval.p	$$⊢P={\mathrm{Base}}_G$$
2		legval.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		legval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		legval.l	$$⊢\underset{\dot{}}{\le }={\le }_{𝒢}\left(G\right)$$
5		legval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
6		legid.a	$$⊢\phi \to A\in P$$
7		legid.b	$$⊢\phi \to B\in P$$
8		legtrd.c	$$⊢\phi \to C\in P$$
9		legtrd.d	$$⊢\phi \to D\in P$$
10		legbtwn.1	$$⊢\phi \to \left(A\in \left(CIB\right)\vee B\in \left(CIA\right)\right)$$
11		legbtwn.2	$$⊢\phi \to \left(C\underset{\dot{}}{-}A\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}B\right)$$
12		simpr	$$⊢\left(\phi \wedge A\in \left(CIB\right)\right)\to A\in \left(CIB\right)$$
13	5	adantr	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to G\in {𝒢}_{\mathrm{Tarski}}$$
14	6	adantr	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to A\in P$$
15	7	adantr	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to B\in P$$
16	8	adantr	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to C\in P$$
17		simpr	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to B\in \left(CIA\right)$$
18	1 2 3 13 16 15 14 17	tgbtwncom	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to B\in \left(AIC\right)$$
19	1 2 3 13 15 16	tgbtwntriv1	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to B\in \left(BIC\right)$$
20	11	adantr	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to \left(C\underset{\dot{}}{-}A\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}B\right)$$
21	1 2 3 4 13 16 15 14 17	btwnleg	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to \left(C\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}A\right)$$
22	1 2 3 4 13 16 14 16 15 20 21	legtri3	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to C\underset{\dot{}}{-}A=C\underset{\dot{}}{-}B$$
23	1 2 3 13 16 14 16 15 22	tgcgrcomlr	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to A\underset{\dot{}}{-}C=B\underset{\dot{}}{-}C$$
24		eqidd	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to B\underset{\dot{}}{-}C=B\underset{\dot{}}{-}C$$
25	1 2 3 13 14 15 16 15 15 16 18 19 23 24	tgcgrsub	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to A\underset{\dot{}}{-}B=B\underset{\dot{}}{-}B$$
26	1 2 3 13 14 15 15 25	axtgcgrid	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to A=B$$
27	26 17	eqeltrd	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to A\in \left(CIA\right)$$
28	26	oveq2d	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to CIA=CIB$$
29	27 28	eleqtrd	$$⊢\left(\phi \wedge B\in \left(CIA\right)\right)\to A\in \left(CIB\right)$$
30	12 29 10	mpjaodan	$$⊢\phi \to A\in \left(CIB\right)$$