Theorem hpgid

Description: The half-plane relation is reflexive. Theorem 9.11 of Schwabhauser p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020)

	Ref	Expression
Hypotheses	hpgid.p	|- P = ( Base ` G )
	hpgid.i	|- I = ( Itv ` G )
	hpgid.l	|- L = ( LineG ` G )
	hpgid.g	|- ( ph -> G e. TarskiG )
	hpgid.d	|- ( ph -> D e. ran L )
	hpgid.a	|- ( ph -> A e. P )
	hpgid.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
	hpgid.1	|- ( ph -> -. A e. D )
Assertion	hpgid	|- ( ph -> A ( ( hpG ` G ) ` D ) A )

Proof

Step	Hyp	Ref	Expression
1		hpgid.p	|- P = ( Base ` G )
2		hpgid.i	|- I = ( Itv ` G )
3		hpgid.l	|- L = ( LineG ` G )
4		hpgid.g	|- ( ph -> G e. TarskiG )
5		hpgid.d	|- ( ph -> D e. ran L )
6		hpgid.a	|- ( ph -> A e. P )
7		hpgid.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
8		hpgid.1	|- ( ph -> -. A e. D )
9		simprr	|- ( ( ph /\ ( c e. P /\ A O c ) ) -> A O c )
10	9 9	jca	|- ( ( ph /\ ( c e. P /\ A O c ) ) -> ( A O c /\ A O c ) )
11	1 2 3 4 5 6 7 8	hpgerlem	|- ( ph -> E. c e. P A O c )
12	10 11	reximddv	|- ( ph -> E. c e. P ( A O c /\ A O c ) )
13	1 2 3 7 4 5 6 6	hpgbr	|- ( ph -> ( A ( ( hpG ` G ) ` D ) A <-> E. c e. P ( A O c /\ A O c ) ) )
14	12 13	mpbird	|- ( ph -> A ( ( hpG ` G ) ` D ) A )