Theorem btwnhl1

Description: Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020)

	Ref	Expression
Hypotheses	ishlg.p	|- P = ( Base ` G )
	ishlg.i	|- I = ( Itv ` G )
	ishlg.k	|- K = ( hlG ` G )
	ishlg.a	|- ( ph -> A e. P )
	ishlg.b	|- ( ph -> B e. P )
	ishlg.c	|- ( ph -> C e. P )
	hlln.1	|- ( ph -> G e. TarskiG )
	hltr.d	|- ( ph -> D e. P )
	btwnhl1.1	|- ( ph -> C e. ( A I B ) )
	btwnhl1.2	|- ( ph -> A =/= B )
	btwnhl1.3	|- ( ph -> C =/= A )
Assertion	btwnhl1	|- ( ph -> C ( K ` A ) B )

Proof

Step	Hyp	Ref	Expression
1		ishlg.p	|- P = ( Base ` G )
2		ishlg.i	|- I = ( Itv ` G )
3		ishlg.k	|- K = ( hlG ` G )
4		ishlg.a	|- ( ph -> A e. P )
5		ishlg.b	|- ( ph -> B e. P )
6		ishlg.c	|- ( ph -> C e. P )
7		hlln.1	|- ( ph -> G e. TarskiG )
8		hltr.d	|- ( ph -> D e. P )
9		btwnhl1.1	|- ( ph -> C e. ( A I B ) )
10		btwnhl1.2	|- ( ph -> A =/= B )
11		btwnhl1.3	|- ( ph -> C =/= A )
12	10	necomd	|- ( ph -> B =/= A )
13	9	orcd	|- ( ph -> ( C e. ( A I B ) \/ B e. ( A I C ) ) )
14	1 2 3 6 5 4 7	ishlg	|- ( ph -> ( C ( K ` A ) B <-> ( C =/= A /\ B =/= A /\ ( C e. ( A I B ) \/ B e. ( A I C ) ) ) ) )
15	11 12 13 14	mpbir3and	|- ( ph -> C ( K ` A ) B )