Theorem tgaas1h

Description: tgaas1 with the constructed point H still explicit: the case split on F = H . Equal, and the side equality is a substitution; distinct, and tgaasne closes the branch. BOTH branches are proved -- assuming F = H would be assuming the theorem. (Contributed by JAG , 12-Aug-2026)

	Ref	Expression
Hypotheses	tgaascgra.p	$$⊢P={\mathrm{Base}}_G$$
	tgaascgra.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tgaascgra.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgaascgra.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	tgaascgra.k	$$⊢K={hl}_{𝒢}\left(G\right)$$
	tgaascgra.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgaascgra.a	$$⊢\phi \to A\in P$$
	tgaascgra.b	$$⊢\phi \to B\in P$$
	tgaascgra.c	$$⊢\phi \to C\in P$$
	tgaascgra.d	$$⊢\phi \to D\in P$$
	tgaascgra.e	$$⊢\phi \to E\in P$$
	tgaascgra.f	$$⊢\phi \to F\in P$$
	tgaascgra.h	$$⊢\phi \to H\in P$$
	tgaascgra.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
	tgaascgra.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
	tgaascgra.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
	tgaascgra.3	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EFD”⟩$$
	tgaascgra.4	$$⊢\phi \to HK\left(E\right)F$$
	tgaascgra.5	$$⊢\phi \to E\underset{\dot{}}{-}H=B\underset{\dot{}}{-}C$$
Assertion	tgaas1h	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$

Proof

Step	Hyp	Ref	Expression
1		tgaascgra.p	$$⊢P={\mathrm{Base}}_G$$
2		tgaascgra.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tgaascgra.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tgaascgra.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
5		tgaascgra.k	$$⊢K={hl}_{𝒢}\left(G\right)$$
6		tgaascgra.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
7		tgaascgra.a	$$⊢\phi \to A\in P$$
8		tgaascgra.b	$$⊢\phi \to B\in P$$
9		tgaascgra.c	$$⊢\phi \to C\in P$$
10		tgaascgra.d	$$⊢\phi \to D\in P$$
11		tgaascgra.e	$$⊢\phi \to E\in P$$
12		tgaascgra.f	$$⊢\phi \to F\in P$$
13		tgaascgra.h	$$⊢\phi \to H\in P$$
14		tgaascgra.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
15		tgaascgra.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
16		tgaascgra.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
17		tgaascgra.3	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EFD”⟩$$
18		tgaascgra.4	$$⊢\phi \to HK\left(E\right)F$$
19		tgaascgra.5	$$⊢\phi \to E\underset{\dot{}}{-}H=B\underset{\dot{}}{-}C$$
20	19	eqcomd	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}H$$
21	20	adantr	$$⊢\left(\phi \wedge F=H\right)\to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}H$$
22		simpr	$$⊢\left(\phi \wedge F=H\right)\to F=H$$
23	22	oveq2d	$$⊢\left(\phi \wedge F=H\right)\to E\underset{\dot{}}{-}F=E\underset{\dot{}}{-}H$$
24	23	eqcomd	$$⊢\left(\phi \wedge F=H\right)\to E\underset{\dot{}}{-}H=E\underset{\dot{}}{-}F$$
25	21 24	eqtrd	$$⊢\left(\phi \wedge F=H\right)\to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
26	6	adantr	$$⊢\left(\phi \wedge F\ne H\right)\to G\in {𝒢}_{\mathrm{Tarski}}$$
27	10	adantr	$$⊢\left(\phi \wedge F\ne H\right)\to D\in P$$
28	11	adantr	$$⊢\left(\phi \wedge F\ne H\right)\to E\in P$$
29	12	adantr	$$⊢\left(\phi \wedge F\ne H\right)\to F\in P$$
30	13	adantr	$$⊢\left(\phi \wedge F\ne H\right)\to H\in P$$
31	1 3 2 6 7 8 9 10 11 12 16 4 14	cgrancol	$$⊢\phi \to \neg \left(F\in \left(DLE\right)\vee D=E\right)$$
32	1 4 3 6 10 11 12 31	ncolrot1	$$⊢\phi \to \neg \left(D\in \left(ELF\right)\vee E=F\right)$$
33	32	adantr	$$⊢\left(\phi \wedge F\ne H\right)\to \neg \left(D\in \left(ELF\right)\vee E=F\right)$$
34	18	adantr	$$⊢\left(\phi \wedge F\ne H\right)\to HK\left(E\right)F$$
35		simpr	$$⊢\left(\phi \wedge F\ne H\right)\to F\ne H$$
36	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19	tgaascgra	$$⊢\phi \to ⟨“DFE”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DHE”⟩$$
37	36	adantr	$$⊢\left(\phi \wedge F\ne H\right)\to ⟨“DFE”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DHE”⟩$$
38	1 2 3 4 5 26 27 28 29 30 33 34 35 37	tgaasne	$$⊢\left(\phi \wedge F\ne H\right)\to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
39	25 38	pm2.61dane	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$