Theorem hlid

Description: The half-line relation is reflexive. Theorem 6.5 of Schwabhauser p. 44. (Contributed by Thierry Arnoux, 21-Feb-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 )
	hlid.1	|- ( ph -> A =/= C )
Assertion	hlid	|- ( ph -> A ( K ` C ) A )

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		hlid.1	|- ( ph -> A =/= C )
9		eqid	|- ( dist ` G ) = ( dist ` G )
10	1 9 2 7 6 4	tgbtwntriv2	|- ( ph -> A e. ( C I A ) )
11	10	olcd	|- ( ph -> ( A e. ( C I A ) \/ A e. ( C I A ) ) )
12	1 2 3 4 4 6 7	ishlg	|- ( ph -> ( A ( K ` C ) A <-> ( A =/= C /\ A =/= C /\ ( A e. ( C I A ) \/ A e. ( C I A ) ) ) ) )
13	8 8 11 12	mpbir3and	|- ( ph -> A ( K ` C ) A )