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Theorem oppne3 25856
Description: Points lying on opposite sides of a line cannot be equal. (Contributed by Thierry Arnoux, 3-Aug-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)
oppcom.a (𝜑𝐴𝑃)
oppcom.b (𝜑𝐵𝑃)
oppcom.o (𝜑𝐴𝑂𝐵)
Assertion
Ref Expression
oppne3 (𝜑𝐴𝐵)
Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐵   𝑡,𝐷   𝑡,𝐺   𝑡,𝐿   𝑡,𝐼   𝑡,𝑂   𝑡,𝑃   𝜑,𝑡   𝑡,   𝑡,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐺(𝑎,𝑏)   𝐿(𝑎,𝑏)   (𝑎,𝑏)   𝑂(𝑎,𝑏)

Proof of Theorem oppne3
StepHypRef Expression
1 hpg.p . . . . . 6 𝑃 = (Base‘𝐺)
2 eqid 2771 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
3 hpg.i . . . . . 6 𝐼 = (Itv‘𝐺)
4 opphl.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 709 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
6 oppcom.a . . . . . . 7 (𝜑𝐴𝑃)
76ad3antrrr 709 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
8 opphl.l . . . . . . 7 𝐿 = (LineG‘𝐺)
9 opphl.d . . . . . . . 8 (𝜑𝐷 ∈ ran 𝐿)
109ad3antrrr 709 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐷 ∈ ran 𝐿)
11 simplr 752 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝐷)
121, 8, 3, 5, 10, 11tglnpt 25665 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝑃)
13 simpr 471 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
14 simpllr 760 . . . . . . . 8 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝐵)
1514oveq2d 6809 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → (𝐴𝐼𝐴) = (𝐴𝐼𝐵))
1613, 15eleqtrrd 2853 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐴))
171, 2, 3, 5, 7, 12, 16axtgbtwnid 25586 . . . . 5 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝑡)
1817, 11eqeltrd 2850 . . . 4 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝐷)
19 oppcom.o . . . . . . 7 (𝜑𝐴𝑂𝐵)
20 hpg.d . . . . . . . 8 = (dist‘𝐺)
21 hpg.o . . . . . . . 8 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
22 oppcom.b . . . . . . . 8 (𝜑𝐵𝑃)
231, 20, 3, 21, 6, 22islnopp 25852 . . . . . . 7 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
2419, 23mpbid 222 . . . . . 6 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
2524simprd 483 . . . . 5 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
2625adantr 466 . . . 4 ((𝜑𝐴 = 𝐵) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
2718, 26r19.29a 3226 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴𝐷)
281, 20, 3, 21, 8, 9, 4, 6, 22, 19oppne1 25854 . . . 4 (𝜑 → ¬ 𝐴𝐷)
2928adantr 466 . . 3 ((𝜑𝐴 = 𝐵) → ¬ 𝐴𝐷)
3027, 29pm2.65da 818 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
3130neqned 2950 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wrex 3062  cdif 3720   class class class wbr 4786  {copab 4846  ran crn 5250  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-cnv 5257  df-dm 5259  df-rn 5260  df-iota 5994  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-trkgb 25569  df-trkg 25573
This theorem is referenced by:  colopp  25882  colhp  25883  trgcopyeulem  25918
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