Theorem tgbtwnouttr

Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 23-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 )
	tgbtwnouttr.1	|- ( ph -> B =/= C )
	tgbtwnouttr.2	|- ( ph -> B e. ( A I C ) )
	tgbtwnouttr.3	|- ( ph -> C e. ( B I D ) )
Assertion	tgbtwnouttr	|- ( ph -> B e. ( A I D ) )

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		tgbtwnouttr.1	|- ( ph -> B =/= C )
10		tgbtwnouttr.2	|- ( ph -> B e. ( A I C ) )
11		tgbtwnouttr.3	|- ( ph -> C e. ( B I D ) )
12	9	necomd	|- ( ph -> C =/= B )
13	1 2 3 4 6 7 8 11	tgbtwncom	|- ( ph -> C e. ( D I B ) )
14	1 2 3 4 5 6 7 10	tgbtwncom	|- ( ph -> B e. ( C I A ) )
15	1 2 3 4 8 7 6 5 12 13 14	tgbtwnouttr2	|- ( ph -> B e. ( D I A ) )
16	1 2 3 4 8 6 5 15	tgbtwncom	|- ( ph -> B e. ( A I D ) )