Theorem tgsegconeq

Description: Two points that satisfy the conclusion of axtgsegcon are identical. Uniqueness portion of Theorem 2.12 of Schwabhauser p. 29. (Contributed by Thierry Arnoux, 23-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	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgcgrextend.a	$$⊢\phi \to A\in P$$
	tgcgrextend.b	$$⊢\phi \to B\in P$$
	tgcgrextend.c	$$⊢\phi \to C\in P$$
	tgcgrextend.d	$$⊢\phi \to D\in P$$
	tgcgrextend.e	$$⊢\phi \to E\in P$$
	tgcgrextend.f	$$⊢\phi \to F\in P$$
	tgsegconeq.1	$$⊢\phi \to D\ne A$$
	tgsegconeq.2	$$⊢\phi \to A\in \left(DIE\right)$$
	tgsegconeq.3	$$⊢\phi \to A\in \left(DIF\right)$$
	tgsegconeq.4	$$⊢\phi \to A\underset{\dot{}}{-}E=B\underset{\dot{}}{-}C$$
	tgsegconeq.5	$$⊢\phi \to A\underset{\dot{}}{-}F=B\underset{\dot{}}{-}C$$
Assertion	tgsegconeq	$$⊢\phi \to E=F$$

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		tgcgrextend.a	$$⊢\phi \to A\in P$$
6		tgcgrextend.b	$$⊢\phi \to B\in P$$
7		tgcgrextend.c	$$⊢\phi \to C\in P$$
8		tgcgrextend.d	$$⊢\phi \to D\in P$$
9		tgcgrextend.e	$$⊢\phi \to E\in P$$
10		tgcgrextend.f	$$⊢\phi \to F\in P$$
11		tgsegconeq.1	$$⊢\phi \to D\ne A$$
12		tgsegconeq.2	$$⊢\phi \to A\in \left(DIE\right)$$
13		tgsegconeq.3	$$⊢\phi \to A\in \left(DIF\right)$$
14		tgsegconeq.4	$$⊢\phi \to A\underset{\dot{}}{-}E=B\underset{\dot{}}{-}C$$
15		tgsegconeq.5	$$⊢\phi \to A\underset{\dot{}}{-}F=B\underset{\dot{}}{-}C$$
16		eqidd	$$⊢\phi \to D\underset{\dot{}}{-}A=D\underset{\dot{}}{-}A$$
17		eqidd	$$⊢\phi \to A\underset{\dot{}}{-}E=A\underset{\dot{}}{-}E$$
18	14 15	eqtr4d	$$⊢\phi \to A\underset{\dot{}}{-}E=A\underset{\dot{}}{-}F$$
19	1 2 3 4 8 5 9 8 5 10 12 13 16 18	tgcgrextend	$$⊢\phi \to D\underset{\dot{}}{-}E=D\underset{\dot{}}{-}F$$
20	1 2 3 4 8 5 9 8 5 9 9 10 11 12 12 16 17 19 18	axtg5seg	$$⊢\phi \to E\underset{\dot{}}{-}E=E\underset{\dot{}}{-}F$$
21	20	eqcomd	$$⊢\phi \to E\underset{\dot{}}{-}F=E\underset{\dot{}}{-}E$$
22	1 2 3 4 9 10 9 21	axtgcgrid	$$⊢\phi \to E=F$$