Theorem opphllem4 25642

Description: Lemma for opphl 25646. (Contributed by Thierry Arnoux, 22-Feb-2020.)

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 )
opphl.k	|- K = (hlG ` G )
opphllem5.n	|- N = ( (pInvG ` G ) ` M )
opphllem5.a	|- ( ph -> A e. P )
opphllem5.c	|- ( ph -> C e. P )
opphllem5.r	|- ( ph -> R e. D )
opphllem5.s	|- ( ph -> S e. D )
opphllem5.m	|- ( ph -> M e. P )
opphllem5.o	|- ( ph -> A O C )
opphllem5.p	|- ( ph -> D (⟂G ` G ) ( A L R ) )
opphllem5.q	|- ( ph -> D (⟂G ` G ) ( C L S ) )
opphllem3.t	|- ( ph -> R =/= S )
opphllem3.l	|- ( ph -> ( S .- C ) (≤G ` G ) ( R .- A ) )
opphllem3.u	|- ( ph -> U e. P )
opphllem3.v	|- ( ph -> ( N ` R ) = S )
opphllem4.u	|- ( ph -> V e. P )
opphllem4.1	|- ( ph -> U ( K ` R ) A )
opphllem4.2	|- ( ph -> V ( K ` S ) C )

Assertion

Ref	Expression
opphllem4	|- ( ph -> U O V )

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

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

Proof of Theorem opphllem4

Step	Hyp	Ref	Expression
1		hpg.p	. 2 |- P = ( Base ` G )
2		hpg.d	. 2 |- .- = ( dist ` G )
3		hpg.i	. 2 |- I = (Itv ` G )
4		hpg.o	. 2 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
5		opphl.l	. 2 |- L = (LineG ` G )
6		opphl.d	. 2 |- ( ph -> D e. ran L )
7		opphl.g	. 2 |- ( ph -> G e. TarskiG )
8		opphllem4.u	. 2 |- ( ph -> V e. P )
9		opphllem3.u	. 2 |- ( ph -> U e. P )
10		opphllem5.n	. . 3 |- N = ( (pInvG ` G ) ` M )
11		eqid 2622	. . . 4 |- (pInvG ` G ) = (pInvG ` G )
12		opphllem5.m	. . . 4 |- ( ph -> M e. P )
13	1, 2, 3, 5, 11, 7, 12, 10, 9	mircl 25556	. . 3 |- ( ph -> ( N ` U ) e. P )
14		opphllem5.s	. . 3 |- ( ph -> S e. D )
15		opphllem5.o	. . . . . . . . . . 11 |- ( ph -> A O C )
16		opphllem5.a	. . . . . . . . . . . 12 |- ( ph -> A e. P )
17		opphllem5.c	. . . . . . . . . . . 12 |- ( ph -> C e. P )
18	1, 2, 3, 4, 16, 17	islnopp 25631	. . . . . . . . . . 11 |- ( ph -> ( A O C <-> ( ( -. A e. D /\ -. C e. D ) /\ E. t e. D t e. ( A I C ) ) ) )
19	15, 18	mpbid 222	. . . . . . . . . 10 |- ( ph -> ( ( -. A e. D /\ -. C e. D ) /\ E. t e. D t e. ( A I C ) ) )
20	19	simpld 475	. . . . . . . . 9 |- ( ph -> ( -. A e. D /\ -. C e. D ) )
21	20	simpld 475	. . . . . . . 8 |- ( ph -> -. A e. D )
22		opphllem5.r	. . . . . . . . . . . 12 |- ( ph -> R e. D )
23	1, 5, 3, 7, 6, 22	tglnpt 25444	. . . . . . . . . . 11 |- ( ph -> R e. P )
24		opphl.k	. . . . . . . . . . . . 13 |- K = (hlG ` G )
25		opphllem4.1	. . . . . . . . . . . . 13 |- ( ph -> U ( K ` R ) A )
26	1, 3, 24, 9, 16, 23, 7, 25	hlne1 25500	. . . . . . . . . . . 12 |- ( ph -> U =/= R )
27	26	necomd 2849	. . . . . . . . . . 11 |- ( ph -> R =/= U )
28	1, 3, 24, 9, 16, 23, 7, 5, 25	hlln 25502	. . . . . . . . . . 11 |- ( ph -> U e. ( A L R ) )
29	1, 3, 24, 9, 16, 23, 7	ishlg 25497	. . . . . . . . . . . . 13 |- ( ph -> ( U ( K ` R ) A <-> ( U =/= R /\ A =/= R /\ ( U e. ( R I A ) \/ A e. ( R I U ) ) ) ) )
30	25, 29	mpbid 222	. . . . . . . . . . . 12 |- ( ph -> ( U =/= R /\ A =/= R /\ ( U e. ( R I A ) \/ A e. ( R I U ) ) ) )
31	30	simp2d 1074	. . . . . . . . . . 11 |- ( ph -> A =/= R )
32	1, 3, 5, 7, 23, 9, 16, 27, 28, 31	lnrot1 25518	. . . . . . . . . 10 |- ( ph -> A e. ( R L U ) )
33	32	adantr 481	. . . . . . . . 9 |- ( ( ph /\ U e. D ) -> A e. ( R L U ) )
34	7	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ U e. D ) -> G e. TarskiG )
35	23	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ U e. D ) -> R e. P )
36	9	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ U e. D ) -> U e. P )
37	27	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ U e. D ) -> R =/= U )
38	6	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ U e. D ) -> D e. ran L )
39	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ U e. D ) -> R e. D )
40		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ U e. D ) -> U e. D )
41	1, 3, 5, 34, 35, 36, 37, 37, 38, 39, 40	tglinethru 25531	. . . . . . . . 9 |- ( ( ph /\ U e. D ) -> D = ( R L U ) )
42	33, 41	eleqtrrd 2704	. . . . . . . 8 |- ( ( ph /\ U e. D ) -> A e. D )
43	21, 42	mtand 691	. . . . . . 7 |- ( ph -> -. U e. D )
44	7	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( N ` U ) e. D ) -> G e. TarskiG )
45	12	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( N ` U ) e. D ) -> M e. P )
46	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( N ` U ) e. D ) -> U e. P )
47	1, 2, 3, 5, 11, 44, 45, 10, 46	mirmir 25557	. . . . . . . 8 |- ( ( ph /\ ( N ` U ) e. D ) -> ( N ` ( N ` U ) ) = U )
48	6	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( N ` U ) e. D ) -> D e. ran L )
49	1, 5, 3, 7, 6, 14	tglnpt 25444	. . . . . . . . . . . 12 |- ( ph -> S e. P )
50		opphllem3.t	. . . . . . . . . . . . 13 |- ( ph -> R =/= S )
51	50	necomd 2849	. . . . . . . . . . . 12 |- ( ph -> S =/= R )
52	1, 2, 3, 5, 11, 7, 12, 10, 23	mirbtwn 25553	. . . . . . . . . . . . 13 |- ( ph -> M e. ( ( N ` R ) I R ) )
53		opphllem3.v	. . . . . . . . . . . . . 14 |- ( ph -> ( N ` R ) = S )
54	53	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ph -> ( ( N ` R ) I R ) = ( S I R ) )
55	52, 54	eleqtrd 2703	. . . . . . . . . . . 12 |- ( ph -> M e. ( S I R ) )
56	1, 3, 5, 7, 49, 23, 12, 51, 55	btwnlng1 25514	. . . . . . . . . . 11 |- ( ph -> M e. ( S L R ) )
57	1, 3, 5, 7, 49, 23, 51, 51, 6, 14, 22	tglinethru 25531	. . . . . . . . . . 11 |- ( ph -> D = ( S L R ) )
58	56, 57	eleqtrrd 2704	. . . . . . . . . 10 |- ( ph -> M e. D )
59	58	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( N ` U ) e. D ) -> M e. D )
60		simpr 477	. . . . . . . . 9 |- ( ( ph /\ ( N ` U ) e. D ) -> ( N ` U ) e. D )
61	1, 2, 3, 5, 11, 44, 10, 48, 59, 60	mirln 25571	. . . . . . . 8 |- ( ( ph /\ ( N ` U ) e. D ) -> ( N ` ( N ` U ) ) e. D )
62	47, 61	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ ( N ` U ) e. D ) -> U e. D )
63	43, 62	mtand 691	. . . . . 6 |- ( ph -> -. ( N ` U ) e. D )
64	63, 43	jca 554	. . . . 5 |- ( ph -> ( -. ( N ` U ) e. D /\ -. U e. D ) )
65	1, 2, 3, 5, 11, 7, 12, 10, 9	mirbtwn 25553	. . . . . 6 |- ( ph -> M e. ( ( N ` U ) I U ) )
66		eleq1 2689	. . . . . . 7 |- ( t = M -> ( t e. ( ( N ` U ) I U ) <-> M e. ( ( N ` U ) I U ) ) )
67	66	rspcev 3309	. . . . . 6 |- ( ( M e. D /\ M e. ( ( N ` U ) I U ) ) -> E. t e. D t e. ( ( N ` U ) I U ) )
68	58, 65, 67	syl2anc 693	. . . . 5 |- ( ph -> E. t e. D t e. ( ( N ` U ) I U ) )
69	64, 68	jca 554	. . . 4 |- ( ph -> ( ( -. ( N ` U ) e. D /\ -. U e. D ) /\ E. t e. D t e. ( ( N ` U ) I U ) ) )
70	1, 2, 3, 4, 13, 9	islnopp 25631	. . . 4 |- ( ph -> ( ( N ` U ) O U <-> ( ( -. ( N ` U ) e. D /\ -. U e. D ) /\ E. t e. D t e. ( ( N ` U ) I U ) ) ) )
71	69, 70	mpbird 247	. . 3 |- ( ph -> ( N ` U ) O U )
72		eqidd 2623	. . 3 |- ( ph -> ( N ` U ) = ( N ` U ) )
73		opphllem5.p	. . . . . . . 8 |- ( ph -> D (⟂G ` G ) ( A L R ) )
74		opphllem5.q	. . . . . . . 8 |- ( ph -> D (⟂G ` G ) ( C L S ) )
75		opphllem3.l	. . . . . . . 8 |- ( ph -> ( S .- C ) (≤G ` G ) ( R .- A ) )
76	1, 2, 3, 4, 5, 6, 7, 24, 10, 16, 17, 22, 14, 12, 15, 73, 74, 50, 75, 9, 53	opphllem3 25641	. . . . . . 7 |- ( ph -> ( U ( K ` R ) A <-> ( N ` U ) ( K ` S ) C ) )
77	25, 76	mpbid 222	. . . . . 6 |- ( ph -> ( N ` U ) ( K ` S ) C )
78		opphllem4.2	. . . . . . 7 |- ( ph -> V ( K ` S ) C )
79	1, 3, 24, 8, 17, 49, 7, 78	hlcomd 25499	. . . . . 6 |- ( ph -> C ( K ` S ) V )
80	1, 3, 24, 13, 17, 8, 7, 49, 77, 79	hltr 25505	. . . . 5 |- ( ph -> ( N ` U ) ( K ` S ) V )
81	1, 3, 24, 13, 8, 49, 7	ishlg 25497	. . . . 5 |- ( ph -> ( ( N ` U ) ( K ` S ) V <-> ( ( N ` U ) =/= S /\ V =/= S /\ ( ( N ` U ) e. ( S I V ) \/ V e. ( S I ( N ` U ) ) ) ) ) )
82	80, 81	mpbid 222	. . . 4 |- ( ph -> ( ( N ` U ) =/= S /\ V =/= S /\ ( ( N ` U ) e. ( S I V ) \/ V e. ( S I ( N ` U ) ) ) ) )
83	82	simp1d 1073	. . 3 |- ( ph -> ( N ` U ) =/= S )
84	82	simp2d 1074	. . 3 |- ( ph -> V =/= S )
85	82	simp3d 1075	. . 3 |- ( ph -> ( ( N ` U ) e. ( S I V ) \/ V e. ( S I ( N ` U ) ) ) )
86	1, 2, 3, 4, 5, 6, 7, 10, 13, 8, 9, 14, 71, 58, 72, 83, 84, 85	opphllem2 25640	. 2 |- ( ph -> V O U )
87	1, 2, 3, 4, 5, 6, 7, 8, 9, 86	oppcom 25636	1 |- ( ph -> U O V )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = 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  ≤Gcleg 25477  hlGchlg 25495  pInvGcmir 25547  ⟂Gcperpg 25590

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-map 7859  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-leg 25478  df-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  opphllem5  25643