Theorem hpgne2 25654

Description: Points on the open half plane cannot lie on its border. (Contributed by Thierry Arnoux, 4-Mar-2020.)

Hypotheses

Ref	Expression
ishpg.p	|- P = ( Base ` G )
ishpg.i	|- I = (Itv ` G )
ishpg.l	|- L = (LineG ` G )
ishpg.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
ishpg.g	|- ( ph -> G e. TarskiG )
ishpg.d	|- ( ph -> D e. ran L )
hpgbr.a	|- ( ph -> A e. P )
hpgbr.b	|- ( ph -> B e. P )
hpgne1.1	|- ( ph -> A ( (hpG ` G ) ` D ) B )

Assertion

Ref	Expression
hpgne2	|- ( ph -> -. B e. D )

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

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

Proof of Theorem hpgne2

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishpg.p	. . 3 |- P = ( Base ` G )
2		eqid 2622	. . 3 |- ( dist ` G ) = ( dist ` G )
3		ishpg.i	. . 3 |- I = (Itv ` G )
4		ishpg.o	. . 3 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
5		ishpg.l	. . 3 |- L = (LineG ` G )
6		ishpg.d	. . . 4 |- ( ph -> D e. ran L )
7	6	ad2antrr 762	. . 3 |- ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> D e. ran L )
8		ishpg.g	. . . 4 |- ( ph -> G e. TarskiG )
9	8	ad2antrr 762	. . 3 |- ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> G e. TarskiG )
10		hpgbr.b	. . . 4 |- ( ph -> B e. P )
11	10	ad2antrr 762	. . 3 |- ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> B e. P )
12		simplr 792	. . 3 |- ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> c e. P )
13		simprr 796	. . 3 |- ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> B O c )
14	1, 2, 3, 4, 5, 7, 9, 11, 12, 13	oppne1 25633	. 2 |- ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> -. B e. D )
15		hpgne1.1	. . 3 |- ( ph -> A ( (hpG ` G ) ` D ) B )
16		hpgbr.a	. . . 4 |- ( ph -> A e. P )
17	1, 3, 5, 4, 8, 6, 16, 10	hpgbr 25652	. . 3 |- ( ph -> ( A ( (hpG ` G ) ` D ) B <-> E. c e. P ( A O c /\ B O c ) ) )
18	15, 17	mpbid 222	. 2 |- ( ph -> E. c e. P ( A O c /\ B O c ) )
19	14, 18	r19.29a 3078	1 |- ( ph -> -. B e. D )

Syntax hints:   -. wn 3   -> wi 4   /\ 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   distcds 15950  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: (None)