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Theorem opphllem4 28773
Description: Lemma for opphl 28777. (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 2735 . . . 4 (pInvG‘𝐺) = (pInvG‘𝐺)
12 opphllem5.m . . . 4 (𝜑𝑀𝑃)
131, 2, 3, 5, 11, 7, 12, 10, 9mircl 28684 . . 3 (𝜑 → (𝑁𝑈) ∈ 𝑃)
14 opphllem5.s . . 3 (𝜑𝑆𝐷)
151, 5, 3, 7, 6, 14tglnpt 28572 . . . . . 6 (𝜑𝑆𝑃)
16 opphllem5.r . . . . . . 7 (𝜑𝑅𝐷)
171, 5, 3, 7, 6, 16tglnpt 28572 . . . . . 6 (𝜑𝑅𝑃)
18 opphllem3.t . . . . . . 7 (𝜑𝑅𝑆)
1918necomd 2994 . . . . . 6 (𝜑𝑆𝑅)
201, 2, 3, 5, 11, 7, 12, 10, 17mirbtwn 28681 . . . . . . 7 (𝜑𝑀 ∈ ((𝑁𝑅)𝐼𝑅))
21 opphllem3.v . . . . . . . 8 (𝜑 → (𝑁𝑅) = 𝑆)
2221oveq1d 7446 . . . . . . 7 (𝜑 → ((𝑁𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
2320, 22eleqtrd 2841 . . . . . 6 (𝜑𝑀 ∈ (𝑆𝐼𝑅))
241, 3, 5, 7, 15, 17, 12, 19, 23btwnlng1 28642 . . . . 5 (𝜑𝑀 ∈ (𝑆𝐿𝑅))
251, 3, 5, 7, 15, 17, 19, 19, 6, 14, 16tglinethru 28659 . . . . 5 (𝜑𝐷 = (𝑆𝐿𝑅))
2624, 25eleqtrrd 2842 . . . 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 28764 . . . . . 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 28628 . . . . . . . . . 10 (𝜑𝑈𝑅)
3433necomd 2994 . . . . . . . . 9 (𝜑𝑅𝑈)
351, 3, 31, 9, 27, 17, 7, 5, 32hlln 28630 . . . . . . . . 9 (𝜑𝑈 ∈ (𝐴𝐿𝑅))
361, 3, 31, 9, 27, 17, 7, 32hlne2 28629 . . . . . . . . 9 (𝜑𝐴𝑅)
371, 3, 5, 7, 17, 9, 27, 34, 35, 36lnrot1 28646 . . . . . . . 8 (𝜑𝐴 ∈ (𝑅𝐿𝑈))
3837adantr 480 . . . . . . 7 ((𝜑𝑈𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
397adantr 480 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝐺 ∈ TarskiG)
4017adantr 480 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑅𝑃)
419adantr 480 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑈𝑃)
4234adantr 480 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑅𝑈)
436adantr 480 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝐷 ∈ ran 𝐿)
4416adantr 480 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑅𝐷)
45 simpr 484 . . . . . . . 8 ((𝜑𝑈𝐷) → 𝑈𝐷)
461, 3, 5, 39, 40, 41, 42, 42, 43, 44, 45tglinethru 28659 . . . . . . 7 ((𝜑𝑈𝐷) → 𝐷 = (𝑅𝐿𝑈))
4738, 46eleqtrrd 2842 . . . . . 6 ((𝜑𝑈𝐷) → 𝐴𝐷)
4830, 47mtand 816 . . . . 5 (𝜑 → ¬ 𝑈𝐷)
497adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
5012adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑀𝑃)
519adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑈𝑃)
521, 2, 3, 5, 11, 49, 50, 10, 51mirmir 28685 . . . . . 6 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁‘(𝑁𝑈)) = 𝑈)
536adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
5426adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑀𝐷)
55 simpr 484 . . . . . . 7 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁𝑈) ∈ 𝐷)
561, 2, 3, 5, 11, 49, 10, 53, 54, 55mirln 28699 . . . . . 6 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → (𝑁‘(𝑁𝑈)) ∈ 𝐷)
5752, 56eqeltrrd 2840 . . . . 5 ((𝜑 ∧ (𝑁𝑈) ∈ 𝐷) → 𝑈𝐷)
5848, 57mtand 816 . . . 4 (𝜑 → ¬ (𝑁𝑈) ∈ 𝐷)
591, 2, 3, 5, 11, 7, 12, 10, 9mirbtwn 28681 . . . 4 (𝜑𝑀 ∈ ((𝑁𝑈)𝐼𝑈))
601, 2, 3, 4, 13, 9, 26, 58, 48, 59islnoppd 28763 . . 3 (𝜑 → (𝑁𝑈)𝑂𝑈)
61 eqidd 2736 . . 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 28772 . . . . . . 7 (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
6632, 65mpbid 232 . . . . . 6 (𝜑 → (𝑁𝑈)(𝐾𝑆)𝐶)
67 opphllem4.2 . . . . . . 7 (𝜑𝑉(𝐾𝑆)𝐶)
681, 3, 31, 8, 28, 15, 7, 67hlcomd 28627 . . . . . 6 (𝜑𝐶(𝐾𝑆)𝑉)
691, 3, 31, 13, 28, 8, 7, 15, 66, 68hltr 28633 . . . . 5 (𝜑 → (𝑁𝑈)(𝐾𝑆)𝑉)
701, 3, 31, 13, 8, 15, 7ishlg 28625 . . . . 5 (𝜑 → ((𝑁𝑈)(𝐾𝑆)𝑉 ↔ ((𝑁𝑈) ≠ 𝑆𝑉𝑆 ∧ ((𝑁𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁𝑈))))))
7169, 70mpbid 232 . . . 4 (𝜑 → ((𝑁𝑈) ≠ 𝑆𝑉𝑆 ∧ ((𝑁𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁𝑈)))))
7271simp1d 1141 . . 3 (𝜑 → (𝑁𝑈) ≠ 𝑆)
731, 3, 31, 28, 8, 15, 7, 68hlne2 28629 . . 3 (𝜑𝑉𝑆)
7471simp3d 1143 . . 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 28771 . 2 (𝜑𝑉𝑂𝑈)
761, 2, 3, 4, 5, 6, 7, 8, 9, 75oppcom 28767 1 (𝜑𝑈𝑂𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  cdif 3960   class class class wbr 5148  {copab 5210  ran crn 5690  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  ≤Gcleg 28605  hlGchlg 28623  pInvGcmir 28675  ⟂Gcperpg 28718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-s3 14885  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476  df-cgrg 28534  df-leg 28606  df-hlg 28624  df-mir 28676  df-rag 28717  df-perpg 28719
This theorem is referenced by:  opphllem5  28774
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