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Theorem opphllem4 26547
Description: Lemma for opphl 26551. (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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
opphl.l 𝐿 = (LineG‘𝐺)
opphl.d (𝜑𝐷 ∈ ran 𝐿)
opphl.g (𝜑𝐺 ∈ TarskiG)
opphl.k 𝐾 = (hlG‘𝐺)
opphllem5.n 𝑁 = ((pInvG‘𝐺)‘𝑀)
opphllem5.a (𝜑𝐴𝑃)
opphllem5.c (𝜑𝐶𝑃)
opphllem5.r (𝜑𝑅𝐷)
opphllem5.s (𝜑𝑆𝐷)
opphllem5.m (𝜑𝑀𝑃)
opphllem5.o (𝜑𝐴𝑂𝐶)
opphllem5.p (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
opphllem5.q (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
opphllem3.t (𝜑𝑅𝑆)
opphllem3.l (𝜑 → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))
opphllem3.u (𝜑𝑈𝑃)
opphllem3.v (𝜑 → (𝑁𝑅) = 𝑆)
opphllem4.u (𝜑𝑉𝑃)
opphllem4.1 (𝜑𝑈(𝐾𝑅)𝐴)
opphllem4.2 (𝜑𝑉(𝐾𝑆)𝐶)
Assertion
Ref Expression
opphllem4 (𝜑𝑈𝑂𝑉)
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
StepHypRef 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5 opphl.l . 2 𝐿 = (LineG‘𝐺)
6 opphl.d . 2 (𝜑𝐷 ∈ ran 𝐿)
7 opphl.g . 2 (𝜑𝐺 ∈ TarskiG)
8 opphllem4.u . 2 (𝜑𝑉𝑃)
9 opphllem3.u . 2 (𝜑𝑈𝑃)
10 opphllem5.n . . 3 𝑁 = ((pInvG‘𝐺)‘𝑀)
11 eqid 2801 . . . 4 (pInvG‘𝐺) = (pInvG‘𝐺)
12 opphllem5.m . . . 4 (𝜑𝑀𝑃)
131, 2, 3, 5, 11, 7, 12, 10, 9mircl 26458 . . 3 (𝜑 → (𝑁𝑈) ∈ 𝑃)
14 opphllem5.s . . 3 (𝜑𝑆𝐷)
151, 5, 3, 7, 6, 14tglnpt 26346 . . . . . 6 (𝜑𝑆𝑃)
16 opphllem5.r . . . . . . 7 (𝜑𝑅𝐷)
171, 5, 3, 7, 6, 16tglnpt 26346 . . . . . 6 (𝜑𝑅𝑃)
18 opphllem3.t . . . . . . 7 (𝜑𝑅𝑆)
1918necomd 3045 . . . . . 6 (𝜑𝑆𝑅)
201, 2, 3, 5, 11, 7, 12, 10, 17mirbtwn 26455 . . . . . . 7 (𝜑𝑀 ∈ ((𝑁𝑅)𝐼𝑅))
21 opphllem3.v . . . . . . . 8 (𝜑 → (𝑁𝑅) = 𝑆)
2221oveq1d 7154 . . . . . . 7 (𝜑 → ((𝑁𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
2320, 22eleqtrd 2895 . . . . . 6 (𝜑𝑀 ∈ (𝑆𝐼𝑅))
241, 3, 5, 7, 15, 17, 12, 19, 23btwnlng1 26416 . . . . 5 (𝜑𝑀 ∈ (𝑆𝐿𝑅))
251, 3, 5, 7, 15, 17, 19, 19, 6, 14, 16tglinethru 26433 . . . . 5 (𝜑𝐷 = (𝑆𝐿𝑅))
2624, 25eleqtrrd 2896 . . . 4 (𝜑𝑀𝐷)
27 opphllem5.a . . . . . . 7 (𝜑𝐴𝑃)
28 opphllem5.c . . . . . . 7 (𝜑𝐶𝑃)
29 opphllem5.o . . . . . . 7 (𝜑𝐴𝑂𝐶)
301, 2, 3, 4, 5, 6, 7, 27, 28, 29oppne1 26538 . . . . . 6 (𝜑 → ¬ 𝐴𝐷)
31 opphl.k . . . . . . . . . . 11 𝐾 = (hlG‘𝐺)
32 opphllem4.1 . . . . . . . . . . 11 (𝜑𝑈(𝐾𝑅)𝐴)
331, 3, 31, 9, 27, 17, 7, 32hlne1 26402 . . . . . . . . . 10 (𝜑𝑈𝑅)
3433necomd 3045 . . . . . . . . 9 (𝜑𝑅𝑈)
351, 3, 31, 9, 27, 17, 7, 5, 32hlln 26404 . . . . . . . . 9 (𝜑𝑈 ∈ (𝐴𝐿𝑅))
361, 3, 31, 9, 27, 17, 7, 32hlne2 26403 . . . . . . . . 9 (𝜑𝐴𝑅)
371, 3, 5, 7, 17, 9, 27, 34, 35, 36lnrot1 26420 . . . . . . . 8 (𝜑𝐴 ∈ (𝑅𝐿𝑈))
3837adantr 484 . . . . . . 7 ((𝜑𝑈𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
397adantr 484 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝐺 ∈ TarskiG)
4017adantr 484 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑅𝑃)
419adantr 484 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑈𝑃)
4234adantr 484 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑅𝑈)
436adantr 484 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝐷 ∈ ran 𝐿)
4416adantr 484 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑅𝐷)
45 simpr 488 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑈𝐷)
461, 3, 5, 39, 40, 41, 42, 42, 43, 44, 45tglinethru 26433 . . . . . . 7 ((𝜑𝑈𝐷) → 𝐷 = (𝑅𝐿𝑈))
4738, 46eleqtrrd 2896 . . . . . 6 ((𝜑𝑈𝐷) → 𝐴𝐷)
4830, 47mtand 815 . . . . 5 (𝜑 → ¬ 𝑈𝐷)
497adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
5012adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑀𝑃)
519adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑈𝑃)
521, 2, 3, 5, 11, 49, 50, 10, 51mirmir 26459 . . . . . 6 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁‘(𝑁𝑈)) = 𝑈)
536adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
5426adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑀𝐷)
55 simpr 488 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁𝑈) ∈ 𝐷)
561, 2, 3, 5, 11, 49, 10, 53, 54, 55mirln 26473 . . . . . 6 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁‘(𝑁𝑈)) ∈ 𝐷)
5752, 56eqeltrrd 2894 . . . . 5 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑈𝐷)
5848, 57mtand 815 . . . 4 (𝜑 → ¬ (𝑁𝑈) ∈ 𝐷)
591, 2, 3, 5, 11, 7, 12, 10, 9mirbtwn 26455 . . . 4 (𝜑𝑀 ∈ ((𝑁𝑈)𝐼𝑈))
601, 2, 3, 4, 13, 9, 26, 58, 48, 59islnoppd 26537 . . 3 (𝜑 → (𝑁𝑈)𝑂𝑈)
61 eqidd 2802 . . 3 (𝜑 → (𝑁𝑈) = (𝑁𝑈))
62 opphllem5.p . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
63 opphllem5.q . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
64 opphllem3.l . . . . . . . 8 (𝜑 → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))
651, 2, 3, 4, 5, 6, 7, 31, 10, 27, 28, 16, 14, 12, 29, 62, 63, 18, 64, 9, 21opphllem3 26546 . . . . . . 7 (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
6632, 65mpbid 235 . . . . . 6 (𝜑 → (𝑁𝑈)(𝐾𝑆)𝐶)
67 opphllem4.2 . . . . . . 7 (𝜑𝑉(𝐾𝑆)𝐶)
681, 3, 31, 8, 28, 15, 7, 67hlcomd 26401 . . . . . 6 (𝜑𝐶(𝐾𝑆)𝑉)
691, 3, 31, 13, 28, 8, 7, 15, 66, 68hltr 26407 . . . . 5 (𝜑 → (𝑁𝑈)(𝐾𝑆)𝑉)
701, 3, 31, 13, 8, 15, 7ishlg 26399 . . . . 5 (𝜑 → ((𝑁𝑈)(𝐾𝑆)𝑉 ↔ ((𝑁𝑈) ≠ 𝑆𝑉𝑆 ∧ ((𝑁𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁𝑈))))))
7169, 70mpbid 235 . . . 4 (𝜑 → ((𝑁𝑈) ≠ 𝑆𝑉𝑆 ∧ ((𝑁𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁𝑈)))))
7271simp1d 1139 . . 3 (𝜑 → (𝑁𝑈) ≠ 𝑆)
731, 3, 31, 28, 8, 15, 7, 68hlne2 26403 . . 3 (𝜑𝑉𝑆)
7471simp3d 1141 . . 3 (𝜑 → ((𝑁𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁𝑈))))
751, 2, 3, 4, 5, 6, 7, 10, 13, 8, 9, 14, 60, 26, 61, 72, 73, 74opphllem2 26545 . 2 (𝜑𝑉𝑂𝑈)
761, 2, 3, 4, 5, 6, 7, 8, 9, 75oppcom 26541 1 (𝜑𝑈𝑂𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wrex 3110  cdif 3881   class class class wbr 5033  {copab 5095  ran crn 5524  cfv 6328  (class class class)co 7139  Basecbs 16478  distcds 16569  TarskiGcstrkg 26227  Itvcitv 26233  LineGclng 26234  ≤Gcleg 26379  hlGchlg 26397  pInvGcmir 26449  ⟂Gcperpg 26492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862  df-concat 13918  df-s1 13945  df-s2 14205  df-s3 14206  df-trkgc 26245  df-trkgb 26246  df-trkgcb 26247  df-trkg 26250  df-cgrg 26308  df-leg 26380  df-hlg 26398  df-mir 26450  df-rag 26491  df-perpg 26493
This theorem is referenced by:  opphllem5  26548
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