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 = ( Base ‘ 𝐺 )
	tgextan.m	⊢ − = ( dist ‘ 𝐺 )
	tgextan.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tgextan.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	tgextan.k	⊢ 𝐾 = ( hlG ‘ 𝐺 )
	tgextan.s	⊢ ≤ = ( ≤∠ ‘ 𝐺 )
	tgextan.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgextan.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgextan.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgextan.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgextan.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	tgextan.1	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
	tgextan.2	⊢ ( 𝜑 → ¬ ( 𝐴 ∈ ( 𝐵 𝐿 𝐶 ) ∨ 𝐵 = 𝐶 ) )
	tgextan.3	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )
	tgextan.cgr	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐴 𝐶 𝐷 ”⟩ )
Assertion	tgextan	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		tgextan.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tgextan.m	⊢ − = ( dist ‘ 𝐺 )
3		tgextan.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tgextan.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
5		tgextan.k	⊢ 𝐾 = ( hlG ‘ 𝐺 )
6		tgextan.s	⊢ ≤ = ( ≤∠ ‘ 𝐺 )
7		tgextan.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
8		tgextan.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
9		tgextan.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
10		tgextan.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
11		tgextan.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
12		tgextan.1	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
13		tgextan.2	⊢ ( 𝜑 → ¬ ( 𝐴 ∈ ( 𝐵 𝐿 𝐶 ) ∨ 𝐵 = 𝐶 ) )
14		tgextan.3	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )
15		tgextan.cgr	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐴 𝐶 𝐷 ”⟩ )
16	1 2 3 4 5 6 7 8 9 10 11 12 13 14	tgexta	⊢ ( 𝜑 → ( ( ⟨“ 𝐵 𝐴 𝐶 ”⟩ ≤ ⟨“ 𝐴 𝐶 𝐷 ”⟩ ∧ ¬ ⟨“ 𝐵 𝐴 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐴 𝐶 𝐷 ”⟩ ) ∧ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ≤ ⟨“ 𝐴 𝐶 𝐷 ”⟩ ∧ ¬ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐴 𝐶 𝐷 ”⟩ ) ) )
17	16	simprd	⊢ ( 𝜑 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ≤ ⟨“ 𝐴 𝐶 𝐷 ”⟩ ∧ ¬ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐴 𝐶 𝐷 ”⟩ ) )
18	17	simprd	⊢ ( 𝜑 → ¬ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐴 𝐶 𝐷 ”⟩ )
19	15 18	pm2.21dd	⊢ ( 𝜑 → 𝜓 )