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Theorem opphllem4 25360
Description: Lemma for opphl 25364. (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 2609 . . . 4 (pInvG‘𝐺) = (pInvG‘𝐺)
12 opphllem5.m . . . 4 (𝜑𝑀𝑃)
131, 2, 3, 5, 11, 7, 12, 10, 9mircl 25274 . . 3 (𝜑 → (𝑁𝑈) ∈ 𝑃)
14 opphllem5.s . . 3 (𝜑𝑆𝐷)
15 opphllem5.o . . . . . . . . . . 11 (𝜑𝐴𝑂𝐶)
16 opphllem5.a . . . . . . . . . . . 12 (𝜑𝐴𝑃)
17 opphllem5.c . . . . . . . . . . . 12 (𝜑𝐶𝑃)
181, 2, 3, 4, 16, 17islnopp 25349 . . . . . . . . . . 11 (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
1915, 18mpbid 220 . . . . . . . . . 10 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
2019simpld 473 . . . . . . . . 9 (𝜑 → (¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷))
2120simpld 473 . . . . . . . 8 (𝜑 → ¬ 𝐴𝐷)
22 opphllem5.r . . . . . . . . . . . 12 (𝜑𝑅𝐷)
231, 5, 3, 7, 6, 22tglnpt 25162 . . . . . . . . . . 11 (𝜑𝑅𝑃)
24 opphl.k . . . . . . . . . . . . 13 𝐾 = (hlG‘𝐺)
25 opphllem4.1 . . . . . . . . . . . . 13 (𝜑𝑈(𝐾𝑅)𝐴)
261, 3, 24, 9, 16, 23, 7, 25hlne1 25218 . . . . . . . . . . . 12 (𝜑𝑈𝑅)
2726necomd 2836 . . . . . . . . . . 11 (𝜑𝑅𝑈)
281, 3, 24, 9, 16, 23, 7, 5, 25hlln 25220 . . . . . . . . . . 11 (𝜑𝑈 ∈ (𝐴𝐿𝑅))
291, 3, 24, 9, 16, 23, 7ishlg 25215 . . . . . . . . . . . . 13 (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑈𝑅𝐴𝑅 ∧ (𝑈 ∈ (𝑅𝐼𝐴) ∨ 𝐴 ∈ (𝑅𝐼𝑈)))))
3025, 29mpbid 220 . . . . . . . . . . . 12 (𝜑 → (𝑈𝑅𝐴𝑅 ∧ (𝑈 ∈ (𝑅𝐼𝐴) ∨ 𝐴 ∈ (𝑅𝐼𝑈))))
3130simp2d 1066 . . . . . . . . . . 11 (𝜑𝐴𝑅)
321, 3, 5, 7, 23, 9, 16, 27, 28, 31lnrot1 25236 . . . . . . . . . 10 (𝜑𝐴 ∈ (𝑅𝐿𝑈))
3332adantr 479 . . . . . . . . 9 ((𝜑𝑈𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
347adantr 479 . . . . . . . . . 10 ((𝜑𝑈𝐷) → 𝐺 ∈ TarskiG)
3523adantr 479 . . . . . . . . . 10 ((𝜑𝑈𝐷) → 𝑅𝑃)
369adantr 479 . . . . . . . . . 10 ((𝜑𝑈𝐷) → 𝑈𝑃)
3727adantr 479 . . . . . . . . . 10 ((𝜑𝑈𝐷) → 𝑅𝑈)
386adantr 479 . . . . . . . . . 10 ((𝜑𝑈𝐷) → 𝐷 ∈ ran 𝐿)
3922adantr 479 . . . . . . . . . 10 ((𝜑𝑈𝐷) → 𝑅𝐷)
40 simpr 475 . . . . . . . . . 10 ((𝜑𝑈𝐷) → 𝑈𝐷)
411, 3, 5, 34, 35, 36, 37, 37, 38, 39, 40tglinethru 25249 . . . . . . . . 9 ((𝜑𝑈𝐷) → 𝐷 = (𝑅𝐿𝑈))
4233, 41eleqtrrd 2690 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝐴𝐷)
4321, 42mtand 688 . . . . . . 7 (𝜑 → ¬ 𝑈𝐷)
447adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
4512adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑀𝑃)
469adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑈𝑃)
471, 2, 3, 5, 11, 44, 45, 10, 46mirmir 25275 . . . . . . . 8 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁‘(𝑁𝑈)) = 𝑈)
486adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
491, 5, 3, 7, 6, 14tglnpt 25162 . . . . . . . . . . . 12 (𝜑𝑆𝑃)
50 opphllem3.t . . . . . . . . . . . . 13 (𝜑𝑅𝑆)
5150necomd 2836 . . . . . . . . . . . 12 (𝜑𝑆𝑅)
521, 2, 3, 5, 11, 7, 12, 10, 23mirbtwn 25271 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ((𝑁𝑅)𝐼𝑅))
53 opphllem3.v . . . . . . . . . . . . . 14 (𝜑 → (𝑁𝑅) = 𝑆)
5453oveq1d 6542 . . . . . . . . . . . . 13 (𝜑 → ((𝑁𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
5552, 54eleqtrd 2689 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (𝑆𝐼𝑅))
561, 3, 5, 7, 49, 23, 12, 51, 55btwnlng1 25232 . . . . . . . . . . 11 (𝜑𝑀 ∈ (𝑆𝐿𝑅))
571, 3, 5, 7, 49, 23, 51, 51, 6, 14, 22tglinethru 25249 . . . . . . . . . . 11 (𝜑𝐷 = (𝑆𝐿𝑅))
5856, 57eleqtrrd 2690 . . . . . . . . . 10 (𝜑𝑀𝐷)
5958adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑀𝐷)
60 simpr 475 . . . . . . . . 9 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁𝑈) ∈ 𝐷)
611, 2, 3, 5, 11, 44, 10, 48, 59, 60mirln 25289 . . . . . . . 8 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁‘(𝑁𝑈)) ∈ 𝐷)
6247, 61eqeltrrd 2688 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑈𝐷)
6343, 62mtand 688 . . . . . 6 (𝜑 → ¬ (𝑁𝑈) ∈ 𝐷)
6463, 43jca 552 . . . . 5 (𝜑 → (¬ (𝑁𝑈) ∈ 𝐷 ∧ ¬ 𝑈𝐷))
651, 2, 3, 5, 11, 7, 12, 10, 9mirbtwn 25271 . . . . . 6 (𝜑𝑀 ∈ ((𝑁𝑈)𝐼𝑈))
66 eleq1 2675 . . . . . . 7 (𝑡 = 𝑀 → (𝑡 ∈ ((𝑁𝑈)𝐼𝑈) ↔ 𝑀 ∈ ((𝑁𝑈)𝐼𝑈)))
6766rspcev 3281 . . . . . 6 ((𝑀𝐷𝑀 ∈ ((𝑁𝑈)𝐼𝑈)) → ∃𝑡𝐷 𝑡 ∈ ((𝑁𝑈)𝐼𝑈))
6858, 65, 67syl2anc 690 . . . . 5 (𝜑 → ∃𝑡𝐷 𝑡 ∈ ((𝑁𝑈)𝐼𝑈))
6964, 68jca 552 . . . 4 (𝜑 → ((¬ (𝑁𝑈) ∈ 𝐷 ∧ ¬ 𝑈𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ ((𝑁𝑈)𝐼𝑈)))
701, 2, 3, 4, 13, 9islnopp 25349 . . . 4 (𝜑 → ((𝑁𝑈)𝑂𝑈 ↔ ((¬ (𝑁𝑈) ∈ 𝐷 ∧ ¬ 𝑈𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ ((𝑁𝑈)𝐼𝑈))))
7169, 70mpbird 245 . . 3 (𝜑 → (𝑁𝑈)𝑂𝑈)
72 eqidd 2610 . . 3 (𝜑 → (𝑁𝑈) = (𝑁𝑈))
73 opphllem5.p . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
74 opphllem5.q . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
75 opphllem3.l . . . . . . . 8 (𝜑 → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))
761, 2, 3, 4, 5, 6, 7, 24, 10, 16, 17, 22, 14, 12, 15, 73, 74, 50, 75, 9, 53opphllem3 25359 . . . . . . 7 (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
7725, 76mpbid 220 . . . . . 6 (𝜑 → (𝑁𝑈)(𝐾𝑆)𝐶)
78 opphllem4.2 . . . . . . 7 (𝜑𝑉(𝐾𝑆)𝐶)
791, 3, 24, 8, 17, 49, 7, 78hlcomd 25217 . . . . . 6 (𝜑𝐶(𝐾𝑆)𝑉)
801, 3, 24, 13, 17, 8, 7, 49, 77, 79hltr 25223 . . . . 5 (𝜑 → (𝑁𝑈)(𝐾𝑆)𝑉)
811, 3, 24, 13, 8, 49, 7ishlg 25215 . . . . 5 (𝜑 → ((𝑁𝑈)(𝐾𝑆)𝑉 ↔ ((𝑁𝑈) ≠ 𝑆𝑉𝑆 ∧ ((𝑁𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁𝑈))))))
8280, 81mpbid 220 . . . 4 (𝜑 → ((𝑁𝑈) ≠ 𝑆𝑉𝑆 ∧ ((𝑁𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁𝑈)))))
8382simp1d 1065 . . 3 (𝜑 → (𝑁𝑈) ≠ 𝑆)
8482simp2d 1066 . . 3 (𝜑𝑉𝑆)
8582simp3d 1067 . . 3 (𝜑 → ((𝑁𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁𝑈))))
861, 2, 3, 4, 5, 6, 7, 10, 13, 8, 9, 14, 71, 58, 72, 83, 84, 85opphllem2 25358 . 2 (𝜑𝑉𝑂𝑈)
871, 2, 3, 4, 5, 6, 7, 8, 9, 86oppcom 25354 1 (𝜑𝑈𝑂𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wrex 2896  cdif 3536   class class class wbr 4577  {copab 4636  ran crn 5029  cfv 5790  (class class class)co 6527  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053  ≤Gcleg 25195  hlGchlg 25213  pInvGcmir 25265  ⟂Gcperpg 25308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-s1 13103  df-s2 13390  df-s3 13391  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkg 25069  df-cgrg 25124  df-leg 25196  df-hlg 25214  df-mir 25266  df-rag 25307  df-perpg 25309
This theorem is referenced by:  opphllem5  25361
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