Theorem tgextan

Description: tgexta 's strictness can be used -- a remote interior angle congruent to the exterior angle is contradictory, so anything follows. This is the shape tgaasexc consumes: that branch holds an angle congruence and must close on it, which the non-strict leA -only form of tgexta could never let it do. (Contributed by JAG, 12-Aug-2026)

	Ref	Expression
Hypotheses	tgextan.p	$$⊢P={\mathrm{Base}}_G$$
	tgextan.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tgextan.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgextan.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	tgextan.k	$$⊢K={hl}_{𝒢}\left(G\right)$$
	tgextan.s	$$⊢\underset{\dot{}}{\le }={\le }_{𝒢}^{\angle }\left(G\right)$$
	tgextan.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgextan.a	$$⊢\phi \to A\in P$$
	tgextan.b	$$⊢\phi \to B\in P$$
	tgextan.c	$$⊢\phi \to C\in P$$
	tgextan.d	$$⊢\phi \to D\in P$$
	tgextan.1	$$⊢\phi \to C\in \left(BID\right)$$
	tgextan.2	$$⊢\phi \to \neg \left(A\in \left(BLC\right)\vee B=C\right)$$
	tgextan.3	$$⊢\phi \to C\ne D$$
	tgextan.cgr	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ACD”⟩$$
Assertion	tgextan	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		tgextan.p	$$⊢P={\mathrm{Base}}_G$$
2		tgextan.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tgextan.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tgextan.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
5		tgextan.k	$$⊢K={hl}_{𝒢}\left(G\right)$$
6		tgextan.s	$$⊢\underset{\dot{}}{\le }={\le }_{𝒢}^{\angle }\left(G\right)$$
7		tgextan.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
8		tgextan.a	$$⊢\phi \to A\in P$$
9		tgextan.b	$$⊢\phi \to B\in P$$
10		tgextan.c	$$⊢\phi \to C\in P$$
11		tgextan.d	$$⊢\phi \to D\in P$$
12		tgextan.1	$$⊢\phi \to C\in \left(BID\right)$$
13		tgextan.2	$$⊢\phi \to \neg \left(A\in \left(BLC\right)\vee B=C\right)$$
14		tgextan.3	$$⊢\phi \to C\ne D$$
15		tgextan.cgr	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ACD”⟩$$
16	1 2 3 4 5 6 7 8 9 10 11 12 13 14	tgexta	$$⊢\phi \to \left(\left(⟨“BAC”⟩\underset{\dot{}}{\le }⟨“ACD”⟩\wedge \neg ⟨“BAC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ACD”⟩\right)\wedge \left(⟨“ABC”⟩\underset{\dot{}}{\le }⟨“ACD”⟩\wedge \neg ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ACD”⟩\right)\right)$$
17	16	simprd	$$⊢\phi \to \left(⟨“ABC”⟩\underset{\dot{}}{\le }⟨“ACD”⟩\wedge \neg ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ACD”⟩\right)$$
18	17	simprd	$$⊢\phi \to \neg ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ACD”⟩$$
19	15 18	pm2.21dd	$$⊢\phi \to \psi$$