Theorem hpgerlem 25657

Description: Lemma for the proof that the half-plane relation is an equivalence relation. Lemma 9.10 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 ) ) }
hpgid.1	|- ( ph -> -. A e. D )

Assertion

Ref	Expression
hpgerlem	|- ( ph -> E. c e. P A O c )

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

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

Proof of Theorem hpgerlem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hpgid.l	. . . 4 |- L = (LineG ` G )
2		hpgid.g	. . . 4 |- ( ph -> G e. TarskiG )
3		hpgid.d	. . . 4 |- ( ph -> D e. ran L )
4	1, 2, 3	tglnne0 25535	. . 3 |- ( ph -> D =/= (/) )
5		n0 3931	. . 3 |- ( D =/= (/) <-> E. x x e. D )
6	4, 5	sylib 208	. 2 |- ( ph -> E. x x e. D )
7		hpgid.p	. . . 4 |- P = ( Base ` G )
8		eqid 2622	. . . 4 |- ( dist ` G ) = ( dist ` G )
9		hpgid.i	. . . 4 |- I = (Itv ` G )
10	2	adantr 481	. . . 4 |- ( ( ph /\ x e. D ) -> G e. TarskiG )
11		hpgid.a	. . . . 5 |- ( ph -> A e. P )
12	11	adantr 481	. . . 4 |- ( ( ph /\ x e. D ) -> A e. P )
13	3	adantr 481	. . . . 5 |- ( ( ph /\ x e. D ) -> D e. ran L )
14		simpr 477	. . . . 5 |- ( ( ph /\ x e. D ) -> x e. D )
15	7, 1, 9, 10, 13, 14	tglnpt 25444	. . . 4 |- ( ( ph /\ x e. D ) -> x e. P )
16	3	adantr 481	. . . . . . 7 |- ( ( ph /\ ( # ` P ) = 1 ) -> D e. ran L )
17	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( # ` P ) = 1 ) -> G e. TarskiG )
18		simpr 477	. . . . . . . 8 |- ( ( ph /\ ( # ` P ) = 1 ) -> ( # ` P ) = 1 )
19	7, 9, 1, 17, 18	tglndim0 25524	. . . . . . 7 |- ( ( ph /\ ( # ` P ) = 1 ) -> -. D e. ran L )
20	16, 19	pm2.65da 600	. . . . . 6 |- ( ph -> -. ( # ` P ) = 1 )
21	7, 11	tgldimor 25397	. . . . . . 7 |- ( ph -> ( ( # ` P ) = 1 \/ 2 <_ ( # ` P ) ) )
22	21	ord 392	. . . . . 6 |- ( ph -> ( -. ( # ` P ) = 1 -> 2 <_ ( # ` P ) ) )
23	20, 22	mpd 15	. . . . 5 |- ( ph -> 2 <_ ( # ` P ) )
24	23	adantr 481	. . . 4 |- ( ( ph /\ x e. D ) -> 2 <_ ( # ` P ) )
25	7, 8, 9, 10, 12, 15, 24	tgbtwndiff 25401	. . 3 |- ( ( ph /\ x e. D ) -> E. c e. P ( x e. ( A I c ) /\ x =/= c ) )
26		hpgid.1	. . . . . . . . 9 |- ( ph -> -. A e. D )
27	26	ad4antr 768	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) -> -. A e. D )
28	10	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> G e. TarskiG )
29	15	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> x e. P )
30		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. D ) /\ c e. P ) -> c e. P )
31	30	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> c e. P )
32	12	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. D ) /\ c e. P ) -> A e. P )
33	32	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> A e. P )
34		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> x =/= c )
35		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) -> x e. ( A I c ) )
36	35	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> x e. ( A I c ) )
37	7, 9, 1, 28, 29, 31, 33, 34, 36	btwnlng2 25515	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> A e. ( x L c ) )
38	13	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> D e. ran L )
39	14	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) -> x e. D )
40	39	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> x e. D )
41		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> c e. D )
42	7, 9, 1, 28, 29, 31, 34, 34, 38, 40, 41	tglinethru 25531	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> D = ( x L c ) )
43	37, 42	eleqtrrd 2704	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) /\ c e. D ) -> A e. D )
44	27, 43	mtand 691	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) -> -. c e. D )
45		eleq1 2689	. . . . . . . . . 10 |- ( t = x -> ( t e. ( A I c ) <-> x e. ( A I c ) ) )
46	45	rspcev 3309	. . . . . . . . 9 |- ( ( x e. D /\ x e. ( A I c ) ) -> E. t e. D t e. ( A I c ) )
47	39, 35, 46	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) -> E. t e. D t e. ( A I c ) )
48	27, 44, 47	jca31 557	. . . . . . 7 |- ( ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ x e. ( A I c ) ) /\ x =/= c ) -> ( ( -. A e. D /\ -. c e. D ) /\ E. t e. D t e. ( A I c ) ) )
49	48	anasss 679	. . . . . 6 |- ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ ( x e. ( A I c ) /\ x =/= c ) ) -> ( ( -. A e. D /\ -. c e. D ) /\ E. t e. D t e. ( A I c ) ) )
50		hpgid.o	. . . . . . . 8 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
51	7, 8, 9, 50, 32, 30	islnopp 25631	. . . . . . 7 |- ( ( ( ph /\ x e. D ) /\ c e. P ) -> ( A O c <-> ( ( -. A e. D /\ -. c e. D ) /\ E. t e. D t e. ( A I c ) ) ) )
52	51	adantr 481	. . . . . 6 |- ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ ( x e. ( A I c ) /\ x =/= c ) ) -> ( A O c <-> ( ( -. A e. D /\ -. c e. D ) /\ E. t e. D t e. ( A I c ) ) ) )
53	49, 52	mpbird 247	. . . . 5 |- ( ( ( ( ph /\ x e. D ) /\ c e. P ) /\ ( x e. ( A I c ) /\ x =/= c ) ) -> A O c )
54	53	ex 450	. . . 4 |- ( ( ( ph /\ x e. D ) /\ c e. P ) -> ( ( x e. ( A I c ) /\ x =/= c ) -> A O c ) )
55	54	reximdva 3017	. . 3 |- ( ( ph /\ x e. D ) -> ( E. c e. P ( x e. ( A I c ) /\ x =/= c ) -> E. c e. P A O c ) )
56	25, 55	mpd 15	. 2 |- ( ( ph /\ x e. D ) -> E. c e. P A O c )
57	6, 56	exlimddv 1863	1 |- ( ph -> E. c e. P A O c )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   (/)c0 3915   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   1c1 9937   <_ cle 10075   2c2 11070   #chash 13117   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406

This theorem is referenced by:  hpgid  25658  lnperpex  25695