Theorem hpgcom 25659

Description: The half-plane relation commutes. Theorem 9.12 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)

Hypotheses

Ref	Expression
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

Ref	Expression
hpgcom	|- ( ph -> B ( (hpG ` G ) ` D ) A )

Distinct variable groups:   t, A   t, B   D, a, b, t   G, a, b, t   I, a, b, t   O, a, b, t   P, a, b, t   ph, t

Allowed substitution hints:   ph( a, b)   A( a, b)   B( a, b)   L( t, a, b)

Proof of Theorem hpgcom

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hpgcom.1	. 2 |- ( ph -> A ( (hpG ` G ) ` D ) B )
2		ancom 466	. . . . 5 |- ( ( A O c /\ B O c ) <-> ( B O c /\ A O c ) )
3	2	a1i 11	. . . 4 |- ( ph -> ( ( A O c /\ B O c ) <-> ( B O c /\ A O c ) ) )
4	3	rexbidv 3052	. . 3 |- ( ph -> ( E. c e. P ( A O c /\ B O c ) <-> E. c e. P ( B O c /\ A O c ) ) )
5		hpgid.p	. . . 4 |- P = ( Base ` G )
6		hpgid.i	. . . 4 |- I = (Itv ` G )
7		hpgid.l	. . . 4 |- L = (LineG ` G )
8		hpgid.o	. . . 4 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
9		hpgid.g	. . . 4 |- ( ph -> G e. TarskiG )
10		hpgid.d	. . . 4 |- ( ph -> D e. ran L )
11		hpgid.a	. . . 4 |- ( ph -> A e. P )
12		hpgcom.b	. . . 4 |- ( ph -> B e. P )
13	5, 6, 7, 8, 9, 10, 11, 12	hpgbr 25652	. . 3 |- ( ph -> ( A ( (hpG ` G ) ` D ) B <-> E. c e. P ( A O c /\ B O c ) ) )
14	5, 6, 7, 8, 9, 10, 12, 11	hpgbr 25652	. . 3 |- ( ph -> ( B ( (hpG ` G ) ` D ) A <-> E. c e. P ( B O c /\ A O c ) ) )
15	4, 13, 14	3bitr4d 300	. 2 |- ( ph -> ( A ( (hpG ` G ) ` D ) B <-> B ( (hpG ` G ) ` D ) A ) )
16	1, 15	mpbid 222	1 |- ( ph -> B ( (hpG ` G ) ` D ) A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  hpGchpg 25649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-hpg 25650

This theorem is referenced by:  trgcopyeulem  25697  tgasa1  25739