Theorem tgbtwntriv2

Description: Betweenness always holds for the second endpoint. Theorem 3.1 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019)

	Ref	Expression
Hypotheses	tkgeom.p	$$⊢P={\mathrm{Base}}_G$$
	tkgeom.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tkgeom.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tkgeom.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgbtwntriv2.1	$$⊢\phi \to A\in P$$
	tgbtwntriv2.2	$$⊢\phi \to B\in P$$
Assertion	tgbtwntriv2	$$⊢\phi \to B\in \left(AIB\right)$$

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	$$⊢P={\mathrm{Base}}_G$$
2		tkgeom.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tkgeom.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tkgeom.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tgbtwntriv2.1	$$⊢\phi \to A\in P$$
6		tgbtwntriv2.2	$$⊢\phi \to B\in P$$
7		simprl	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge \left(B\in \left(AIx\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\right)\to B\in \left(AIx\right)$$
8	4	ad2antrr	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\to G\in {𝒢}_{\mathrm{Tarski}}$$
9	6	ad2antrr	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\to B\in P$$
10		simplr	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\to x\in P$$
11		simpr	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\to B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B$$
12	1 2 3 8 9 10 9 11	axtgcgrid	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\to B=x$$
13	12	adantrl	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge \left(B\in \left(AIx\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\right)\to B=x$$
14	13	oveq2d	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge \left(B\in \left(AIx\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\right)\to AIB=AIx$$
15	7 14	eleqtrrd	$$⊢\left(\left(\phi \wedge x\in P\right)\wedge \left(B\in \left(AIx\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)\right)\to B\in \left(AIB\right)$$
16	1 2 3 4 5 6 6 6	axtgsegcon	$$⊢\phi \to \exists x\in P\left(B\in \left(AIx\right)\wedge B\underset{\dot{}}{-}x=B\underset{\dot{}}{-}B\right)$$
17	15 16	r19.29a	$$⊢\phi \to B\in \left(AIB\right)$$