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	|- .<_ = ( leA ` G )
	tgextan.g	|- ( ph -> G e. TarskiG )
	tgextan.a	|- ( ph -> A e. P )
	tgextan.b	|- ( ph -> B e. P )
	tgextan.c	|- ( ph -> C e. P )
	tgextan.d	|- ( ph -> D e. P )
	tgextan.1	|- ( ph -> C e. ( B I D ) )
	tgextan.2	|- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
	tgextan.3	|- ( ph -> C =/= D )
	tgextan.cgr	|- ( ph -> <" A B C "> ( cgrA ` G ) <" A C D "> )
Assertion	tgextan	|- ( ph -> ps )

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	|- .<_ = ( leA ` G )
7		tgextan.g	|- ( ph -> G e. TarskiG )
8		tgextan.a	|- ( ph -> A e. P )
9		tgextan.b	|- ( ph -> B e. P )
10		tgextan.c	|- ( ph -> C e. P )
11		tgextan.d	|- ( ph -> D e. P )
12		tgextan.1	|- ( ph -> C e. ( B I D ) )
13		tgextan.2	|- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
14		tgextan.3	|- ( ph -> C =/= D )
15		tgextan.cgr	|- ( ph -> <" A B C "> ( cgrA ` G ) <" A C D "> )
16	1 2 3 4 5 6 7 8 9 10 11 12 13 14	tgexta	|- ( ph -> ( ( <" 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	|- ( ph -> ( <" A B C "> .<_ <" A C D "> /\ -. <" A B C "> ( cgrA ` G ) <" A C D "> ) )
18	17	simprd	|- ( ph -> -. <" A B C "> ( cgrA ` G ) <" A C D "> )
19	15 18	pm2.21dd	|- ( ph -> ps )