Theorem hpgcom

Description: The half-plane relation commutes. Theorem 9.12 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 ) ) }
	hpgcom.b	|- ( ph -> B e. P )
	hpgcom.1	|- ( ph -> A ( ( hpG ` G ) ` D ) B )
Assertion	hpgcom	|- ( ph -> B ( ( 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		hpgcom.b	|- ( ph -> B e. P )
9		hpgcom.1	|- ( ph -> A ( ( hpG ` G ) ` D ) B )
10		ancom	|- ( ( A O c /\ B O c ) <-> ( B O c /\ A O c ) )
11	10	a1i	|- ( ph -> ( ( A O c /\ B O c ) <-> ( B O c /\ A O c ) ) )
12	11	rexbidv	|- ( ph -> ( E. c e. P ( A O c /\ B O c ) <-> E. c e. P ( B O c /\ A O c ) ) )
13	1 2 3 7 4 5 6 8	hpgbr	|- ( ph -> ( A ( ( hpG ` G ) ` D ) B <-> E. c e. P ( A O c /\ B O c ) ) )
14	1 2 3 7 4 5 8 6	hpgbr	|- ( ph -> ( B ( ( hpG ` G ) ` D ) A <-> E. c e. P ( B O c /\ A O c ) ) )
15	12 13 14	3bitr4d	|- ( ph -> ( A ( ( hpG ` G ) ` D ) B <-> B ( ( hpG ` G ) ` D ) A ) )
16	9 15	mpbid	|- ( ph -> B ( ( hpG ` G ) ` D ) A )