Theorem tgbtwnintr

Description: Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 18-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 )
	tgbtwnintr.1	|- ( ph -> A e. P )
	tgbtwnintr.2	|- ( ph -> B e. P )
	tgbtwnintr.3	|- ( ph -> C e. P )
	tgbtwnintr.4	|- ( ph -> D e. P )
	tgbtwnintr.5	|- ( ph -> A e. ( B I D ) )
	tgbtwnintr.6	|- ( ph -> B e. ( C I D ) )
Assertion	tgbtwnintr	|- ( ph -> B e. ( A I C ) )

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		tgbtwnintr.1	|- ( ph -> A e. P )
6		tgbtwnintr.2	|- ( ph -> B e. P )
7		tgbtwnintr.3	|- ( ph -> C e. P )
8		tgbtwnintr.4	|- ( ph -> D e. P )
9		tgbtwnintr.5	|- ( ph -> A e. ( B I D ) )
10		tgbtwnintr.6	|- ( ph -> B e. ( C I D ) )
11	4	ad2antrr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> G e. TarskiG )
12	6	ad2antrr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> B e. P )
13		simplr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> x e. P )
14		simprr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> x e. ( B I B ) )
15	1 2 3 11 12 13 14	axtgbtwnid	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> B = x )
16		simprl	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> x e. ( A I C ) )
17	15 16	eqeltrd	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> B e. ( A I C ) )
18	1 2 3 4 6 7 8 5 6 9 10	axtgpasch	|- ( ph -> E. x e. P ( x e. ( A I C ) /\ x e. ( B I B ) ) )
19	17 18	r19.29a	|- ( ph -> B e. ( A I C ) )