Theorem tgbtwncom

Description: Betweenness commutes. Theorem 3.2 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgbtwntriv2.1	|- ( ph -> A e. P )
	tgbtwntriv2.2	|- ( ph -> B e. P )
	tgbtwncom.3	|- ( ph -> C e. P )
	tgbtwncom.4	|- ( ph -> B e. ( A I C ) )
Assertion	tgbtwncom	|- ( ph -> B e. ( C I A ) )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	|- P = ( Base ` G )
2		tkgeom.d	|- .- = ( dist ` G )
3		tkgeom.i	|- I = ( Itv ` G )
4		tkgeom.g	|- ( ph -> G e. TarskiG )
5		tgbtwntriv2.1	|- ( ph -> A e. P )
6		tgbtwntriv2.2	|- ( ph -> B e. P )
7		tgbtwncom.3	|- ( ph -> C e. P )
8		tgbtwncom.4	|- ( ph -> B e. ( A I C ) )
9	4	ad2antrr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( B I B ) /\ x e. ( C I A ) ) ) -> G e. TarskiG )
10	6	ad2antrr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( B I B ) /\ x e. ( C I A ) ) ) -> B e. P )
11		simplr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( B I B ) /\ x e. ( C I A ) ) ) -> x e. P )
12		simprl	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( B I B ) /\ x e. ( C I A ) ) ) -> x e. ( B I B ) )
13	1 2 3 9 10 11 12	axtgbtwnid	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( B I B ) /\ x e. ( C I A ) ) ) -> B = x )
14		simprr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( B I B ) /\ x e. ( C I A ) ) ) -> x e. ( C I A ) )
15	13 14	eqeltrd	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( B I B ) /\ x e. ( C I A ) ) ) -> B e. ( C I A ) )
16	1 2 3 4 6 7	tgbtwntriv2	|- ( ph -> C e. ( B I C ) )
17	1 2 3 4 5 6 7 6 7 8 16	axtgpasch	|- ( ph -> E. x e. P ( x e. ( B I B ) /\ x e. ( C I A ) ) )
18	15 17	r19.29a	|- ( ph -> B e. ( C I A ) )