Theorem opphllem2 25640

Description: Lemma for opphl 25646. Lemma 9.3 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 21-Dec-2019.)

Hypotheses

Ref	Expression
hpg.p	|- P = ( Base ` G )
hpg.d	|- .- = ( dist ` G )
hpg.i	|- I = (Itv ` G )
hpg.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
opphl.l	|- L = (LineG ` G )
opphl.d	|- ( ph -> D e. ran L )
opphl.g	|- ( ph -> G e. TarskiG )
opphllem1.s	|- S = ( (pInvG ` G ) ` M )
opphllem1.a	|- ( ph -> A e. P )
opphllem1.b	|- ( ph -> B e. P )
opphllem1.c	|- ( ph -> C e. P )
opphllem1.r	|- ( ph -> R e. D )
opphllem1.o	|- ( ph -> A O C )
opphllem1.m	|- ( ph -> M e. D )
opphllem1.n	|- ( ph -> A = ( S ` C ) )
opphllem1.x	|- ( ph -> A =/= R )
opphllem1.y	|- ( ph -> B =/= R )
opphllem2.z	|- ( ph -> ( A e. ( R I B ) \/ B e. ( R I A ) ) )

Assertion

Ref	Expression
opphllem2	|- ( ph -> B O C )

Distinct variable groups:   D, a, b   I, a, b   P, a, b   t, A   t, B   t, D   t, R   t, C   t, G   t, L   t, I   t, M   t, O   t, P   t, S   ph, t   t, .-   t, a, b

Allowed substitution hints:   ph( a, b)   A( a, b)   B( a, b)   C( a, b)   R( a, b)   S( a, b)   G( a, b)   L( a, b)   M( a, b)   .- ( a, b)   O( a, b)

Proof of Theorem opphllem2

Step	Hyp	Ref	Expression
1		hpg.p	. . 3 |- P = ( Base ` G )
2		hpg.d	. . 3 |- .- = ( dist ` G )
3		hpg.i	. . 3 |- I = (Itv ` G )
4		hpg.o	. . 3 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
5		opphl.l	. . 3 |- L = (LineG ` G )
6		opphl.d	. . . 4 |- ( ph -> D e. ran L )
7	6	adantr 481	. . 3 |- ( ( ph /\ A e. ( R I B ) ) -> D e. ran L )
8		opphl.g	. . . 4 |- ( ph -> G e. TarskiG )
9	8	adantr 481	. . 3 |- ( ( ph /\ A e. ( R I B ) ) -> G e. TarskiG )
10		opphllem1.c	. . . 4 |- ( ph -> C e. P )
11	10	adantr 481	. . 3 |- ( ( ph /\ A e. ( R I B ) ) -> C e. P )
12		opphllem1.b	. . . 4 |- ( ph -> B e. P )
13	12	adantr 481	. . 3 |- ( ( ph /\ A e. ( R I B ) ) -> B e. P )
14		opphllem1.s	. . . 4 |- S = ( (pInvG ` G ) ` M )
15		eqid 2622	. . . . 5 |- (pInvG ` G ) = (pInvG ` G )
16		opphllem1.m	. . . . . . 7 |- ( ph -> M e. D )
17	1, 5, 3, 8, 6, 16	tglnpt 25444	. . . . . 6 |- ( ph -> M e. P )
18	17	adantr 481	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> M e. P )
19	1, 2, 3, 5, 15, 9, 18, 14, 13	mircl 25556	. . . 4 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` B ) e. P )
20	16	adantr 481	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> M e. D )
21		opphllem1.r	. . . . . 6 |- ( ph -> R e. D )
22	21	adantr 481	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> R e. D )
23	1, 2, 3, 5, 15, 9, 14, 7, 20, 22	mirln 25571	. . . 4 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` R ) e. D )
24		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A = B ) -> A = B )
25		simplr 792	. . . . . . . . 9 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A = B ) -> B e. D )
26	24, 25	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A = B ) -> A e. D )
27	8	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> G e. TarskiG )
28	12	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> B e. P )
29	1, 5, 3, 8, 6, 21	tglnpt 25444	. . . . . . . . . . 11 |- ( ph -> R e. P )
30	29	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> R e. P )
31		opphllem1.a	. . . . . . . . . . 11 |- ( ph -> A e. P )
32	31	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> A e. P )
33		opphllem1.y	. . . . . . . . . . 11 |- ( ph -> B =/= R )
34	33	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> B =/= R )
35	34	necomd 2849	. . . . . . . . . . 11 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> R =/= B )
36		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> A e. ( R I B ) )
37	1, 3, 5, 27, 30, 28, 32, 35, 36	btwnlng1 25514	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> A e. ( R L B ) )
38	1, 3, 5, 27, 28, 30, 32, 34, 37	lncom 25517	. . . . . . . . 9 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> A e. ( B L R ) )
39	6	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> D e. ran L )
40		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> B e. D )
41	21	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> R e. D )
42	1, 3, 5, 27, 28, 30, 34, 34, 39, 40, 41	tglinethru 25531	. . . . . . . . 9 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> D = ( B L R ) )
43	38, 42	eleqtrrd 2704	. . . . . . . 8 |- ( ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) /\ A =/= B ) -> A e. D )
44	26, 43	pm2.61dane 2881	. . . . . . 7 |- ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) -> A e. D )
45		opphllem1.o	. . . . . . . . 9 |- ( ph -> A O C )
46	1, 2, 3, 4, 5, 6, 8, 31, 10, 45	oppne1 25633	. . . . . . . 8 |- ( ph -> -. A e. D )
47	46	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ A e. ( R I B ) ) /\ B e. D ) -> -. A e. D )
48	44, 47	pm2.65da 600	. . . . . 6 |- ( ( ph /\ A e. ( R I B ) ) -> -. B e. D )
49	9	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> G e. TarskiG )
50	18	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> M e. P )
51	13	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> B e. P )
52	1, 2, 3, 5, 15, 49, 50, 14, 51	mirmir 25557	. . . . . . 7 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> ( S ` ( S ` B ) ) = B )
53	7	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> D e. ran L )
54	20	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> M e. D )
55		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> ( S ` B ) e. D )
56	1, 2, 3, 5, 15, 49, 14, 53, 54, 55	mirln 25571	. . . . . . 7 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> ( S ` ( S ` B ) ) e. D )
57	52, 56	eqeltrrd 2702	. . . . . 6 |- ( ( ( ph /\ A e. ( R I B ) ) /\ ( S ` B ) e. D ) -> B e. D )
58	48, 57	mtand 691	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> -. ( S ` B ) e. D )
59	1, 2, 3, 5, 15, 9, 18, 14, 13	mirbtwn 25553	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> M e. ( ( S ` B ) I B ) )
60	1, 2, 3, 4, 19, 13, 20, 58, 48, 59	islnoppd 25632	. . . 4 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` B ) O B )
61		eqidd 2623	. . . 4 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` B ) = ( S ` B ) )
62		nelne2 2891	. . . . . 6 |- ( ( ( S ` R ) e. D /\ -. ( S ` B ) e. D ) -> ( S ` R ) =/= ( S ` B ) )
63	23, 58, 62	syl2anc 693	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` R ) =/= ( S ` B ) )
64	63	necomd 2849	. . . 4 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` B ) =/= ( S ` R ) )
65	1, 2, 3, 4, 5, 6, 8, 31, 10, 45	oppne2 25634	. . . . . . 7 |- ( ph -> -. C e. D )
66	65	adantr 481	. . . . . 6 |- ( ( ph /\ A e. ( R I B ) ) -> -. C e. D )
67		nelne2 2891	. . . . . 6 |- ( ( ( S ` R ) e. D /\ -. C e. D ) -> ( S ` R ) =/= C )
68	23, 66, 67	syl2anc 693	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` R ) =/= C )
69	68	necomd 2849	. . . 4 |- ( ( ph /\ A e. ( R I B ) ) -> C =/= ( S ` R ) )
70		opphllem1.n	. . . . . . . 8 |- ( ph -> A = ( S ` C ) )
71	70	eqcomd 2628	. . . . . . 7 |- ( ph -> ( S ` C ) = A )
72	1, 2, 3, 5, 15, 8, 17, 14, 10, 71	mircom 25558	. . . . . 6 |- ( ph -> ( S ` A ) = C )
73	72	adantr 481	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` A ) = C )
74	29	adantr 481	. . . . . 6 |- ( ( ph /\ A e. ( R I B ) ) -> R e. P )
75	31	adantr 481	. . . . . 6 |- ( ( ph /\ A e. ( R I B ) ) -> A e. P )
76		simpr 477	. . . . . 6 |- ( ( ph /\ A e. ( R I B ) ) -> A e. ( R I B ) )
77	1, 2, 3, 5, 15, 9, 18, 14, 74, 75, 13, 76	mirbtwni 25566	. . . . 5 |- ( ( ph /\ A e. ( R I B ) ) -> ( S ` A ) e. ( ( S ` R ) I ( S ` B ) ) )
78	73, 77	eqeltrrd 2702	. . . 4 |- ( ( ph /\ A e. ( R I B ) ) -> C e. ( ( S ` R ) I ( S ` B ) ) )
79	1, 2, 3, 4, 5, 7, 9, 14, 19, 11, 13, 23, 60, 20, 61, 64, 69, 78	opphllem1 25639	. . 3 |- ( ( ph /\ A e. ( R I B ) ) -> C O B )
80	1, 2, 3, 4, 5, 7, 9, 11, 13, 79	oppcom 25636	. 2 |- ( ( ph /\ A e. ( R I B ) ) -> B O C )
81	6	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> D e. ran L )
82	8	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> G e. TarskiG )
83	31	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> A e. P )
84	12	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> B e. P )
85	10	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> C e. P )
86	21	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> R e. D )
87	45	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> A O C )
88	16	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> M e. D )
89	70	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> A = ( S ` C ) )
90		opphllem1.x	. . . 4 |- ( ph -> A =/= R )
91	90	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> A =/= R )
92	33	adantr 481	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> B =/= R )
93		simpr 477	. . 3 |- ( ( ph /\ B e. ( R I A ) ) -> B e. ( R I A ) )
94	1, 2, 3, 4, 5, 81, 82, 14, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93	opphllem1 25639	. 2 |- ( ( ph /\ B e. ( R I A ) ) -> B O C )
95		opphllem2.z	. 2 |- ( ph -> ( A e. ( R I B ) \/ B e. ( R I A ) ) )
96	80, 94, 95	mpjaodan 827	1 |- ( ph -> B O C )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   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  pInvGcmir 25547

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-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  df-mir 25548

This theorem is referenced by:  opphllem4  25642  opphl  25646