Theorem opphllem4 26813

Description: Lemma for opphl 26817. (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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
opphl.l	⊢ 𝐿 = (LineG‘𝐺)
opphl.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
opphl.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
opphl.k	⊢ 𝐾 = (hlG‘𝐺)
opphllem5.n	⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)
opphllem5.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
opphllem5.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
opphllem5.r	⊢ (𝜑 → 𝑅 ∈ 𝐷)
opphllem5.s	⊢ (𝜑 → 𝑆 ∈ 𝐷)
opphllem5.m	⊢ (𝜑 → 𝑀 ∈ 𝑃)
opphllem5.o	⊢ (𝜑 → 𝐴𝑂𝐶)
opphllem5.p	⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
opphllem5.q	⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
opphllem3.t	⊢ (𝜑 → 𝑅 ≠ 𝑆)
opphllem3.l	⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
opphllem3.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
opphllem3.v	⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
opphllem4.u	⊢ (𝜑 → 𝑉 ∈ 𝑃)
opphllem4.1	⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
opphllem4.2	⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)

Assertion

Ref	Expression
opphllem4	⊢ (𝜑 → 𝑈𝑂𝑉)

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   𝜑,𝑡   𝑡, −   𝑡,𝑎,𝑏

Allowed substitution hints:   𝜑(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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5		opphl.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
6		opphl.d	. 2 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
7		opphl.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8		opphllem4.u	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
9		opphllem3.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
10		opphllem5.n	. . 3 ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)
11		eqid 2734	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
12		opphllem5.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
13	1, 2, 3, 5, 11, 7, 12, 10, 9	mircl 26724	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ∈ 𝑃)
14		opphllem5.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐷)
15	1, 5, 3, 7, 6, 14	tglnpt 26612	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
16		opphllem5.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
17	1, 5, 3, 7, 6, 16	tglnpt 26612	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
18		opphllem3.t	. . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ 𝑆)
19	18	necomd 2990	. . . . . 6 ⊢ (𝜑 → 𝑆 ≠ 𝑅)
20	1, 2, 3, 5, 11, 7, 12, 10, 17	mirbtwn 26721	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑅)𝐼𝑅))
21		opphllem3.v	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
22	21	oveq1d 7217	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
23	20, 22	eleqtrd 2836	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐼𝑅))
24	1, 3, 5, 7, 15, 17, 12, 19, 23	btwnlng1 26682	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐿𝑅))
25	1, 3, 5, 7, 15, 17, 19, 19, 6, 14, 16	tglinethru 26699	. . . . 5 ⊢ (𝜑 → 𝐷 = (𝑆𝐿𝑅))
26	24, 25	eleqtrrd 2837	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝐷)
27		opphllem5.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
28		opphllem5.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
29		opphllem5.o	. . . . . . 7 ⊢ (𝜑 → 𝐴𝑂𝐶)
30	1, 2, 3, 4, 5, 6, 7, 27, 28, 29	oppne1 26804	. . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
31		opphl.k	. . . . . . . . . . 11 ⊢ 𝐾 = (hlG‘𝐺)
32		opphllem4.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
33	1, 3, 31, 9, 27, 17, 7, 32	hlne1 26668	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≠ 𝑅)
34	33	necomd 2990	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ≠ 𝑈)
35	1, 3, 31, 9, 27, 17, 7, 5, 32	hlln 26670	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (𝐴𝐿𝑅))
36	1, 3, 31, 9, 27, 17, 7, 32	hlne2 26669	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
37	1, 3, 5, 7, 17, 9, 27, 34, 35, 36	lnrot1 26686	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑅𝐿𝑈))
38	37	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
39	7	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐺 ∈ TarskiG)
40	17	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝑃)
41	9	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝑃)
42	34	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ≠ 𝑈)
43	6	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
44	16	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝐷)
45		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝐷)
46	1, 3, 5, 39, 40, 41, 42, 42, 43, 44, 45	tglinethru 26699	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 = (𝑅𝐿𝑈))
47	38, 46	eleqtrrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ 𝐷)
48	30, 47	mtand 816	. . . . 5 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
49	7	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
50	12	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝑃)
51	9	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝑃)
52	1, 2, 3, 5, 11, 49, 50, 10, 51	mirmir 26725	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
53	6	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
54	26	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝐷)
55		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘𝑈) ∈ 𝐷)
56	1, 2, 3, 5, 11, 49, 10, 53, 54, 55	mirln 26739	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) ∈ 𝐷)
57	52, 56	eqeltrrd 2835	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝐷)
58	48, 57	mtand 816	. . . 4 ⊢ (𝜑 → ¬ (𝑁‘𝑈) ∈ 𝐷)
59	1, 2, 3, 5, 11, 7, 12, 10, 9	mirbtwn 26721	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈))
60	1, 2, 3, 4, 13, 9, 26, 58, 48, 59	islnoppd 26803	. . 3 ⊢ (𝜑 → (𝑁‘𝑈)𝑂𝑈)
61		eqidd 2735	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) = (𝑁‘𝑈))
62		opphllem5.p	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
63		opphllem5.q	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
64		opphllem3.l	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
65	1, 2, 3, 4, 5, 6, 7, 31, 10, 27, 28, 16, 14, 12, 29, 62, 63, 18, 64, 9, 21	opphllem3 26812	. . . . . . 7 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
66	32, 65	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝐶)
67		opphllem4.2	. . . . . . 7 ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)
68	1, 3, 31, 8, 28, 15, 7, 67	hlcomd 26667	. . . . . 6 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
69	1, 3, 31, 13, 28, 8, 7, 15, 66, 68	hltr 26673	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝑉)
70	1, 3, 31, 13, 8, 15, 7	ishlg 26665	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝑈)(𝐾‘𝑆)𝑉 ↔ ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))))
71	69, 70	mpbid 235	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈)))))
72	71	simp1d 1144	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ≠ 𝑆)
73	1, 3, 31, 28, 8, 15, 7, 68	hlne2 26669	. . 3 ⊢ (𝜑 → 𝑉 ≠ 𝑆)
74	71	simp3d 1146	. . 3 ⊢ (𝜑 → ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))
75	1, 2, 3, 4, 5, 6, 7, 10, 13, 8, 9, 14, 60, 26, 61, 72, 73, 74	opphllem2 26811	. 2 ⊢ (𝜑 → 𝑉𝑂𝑈)
76	1, 2, 3, 4, 5, 6, 7, 8, 9, 75	oppcom 26807	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2935  ∃wrex 3055   ∖ cdif 3854   class class class wbr 5043  {copab 5105  ran crn 5541  ‘cfv 6369  (class class class)co 7202  Basecbs 16684  distcds 16776  TarskiGcstrkg 26493  Itvcitv 26499  LineGclng 26500  ≤Gcleg 26645  hlGchlg 26663  pInvGcmir 26715  ⟂Gcperpg 26758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-oadd 8195  df-er 8380  df-map 8499  df-pm 8500  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-dju 9500  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-2 11876  df-3 11877  df-n0 12074  df-xnn0 12146  df-z 12160  df-uz 12422  df-fz 13079  df-fzo 13222  df-hash 13880  df-word 14053  df-concat 14109  df-s1 14136  df-s2 14396  df-s3 14397  df-trkgc 26511  df-trkgb 26512  df-trkgcb 26513  df-trkg 26516  df-cgrg 26574  df-leg 26646  df-hlg 26664  df-mir 26716  df-rag 26757  df-perpg 26759

This theorem is referenced by:  opphllem5  26814