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Theorem oppnid 25438
 Description: The "opposite to a line" relation is irreflexive. (Contributed by Thierry Arnoux, 4-Mar-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)
oppnid.1 (𝜑𝐴𝑃)
Assertion
Ref Expression
oppnid (𝜑 → ¬ 𝐴𝑂𝐴)
Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   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)   𝐺(𝑎,𝑏)   𝐿(𝑎,𝑏)   (𝑎,𝑏)   𝑂(𝑎,𝑏)

Proof of Theorem oppnid
StepHypRef Expression
1 hpg.p . . . . 5 𝑃 = (Base‘𝐺)
2 hpg.d . . . . 5 = (dist‘𝐺)
3 hpg.i . . . . 5 𝐼 = (Itv‘𝐺)
4 opphl.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 762 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐺 ∈ TarskiG)
6 oppnid.1 . . . . . 6 (𝜑𝐴𝑃)
76ad3antrrr 762 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴𝑃)
8 opphl.l . . . . . 6 𝐿 = (LineG‘𝐺)
9 opphl.d . . . . . . 7 (𝜑𝐷 ∈ ran 𝐿)
109ad3antrrr 762 . . . . . 6 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐷 ∈ ran 𝐿)
11 simplr 788 . . . . . 6 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡𝐷)
121, 8, 3, 5, 10, 11tglnpt 25244 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡𝑃)
13 simpr 476 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴))
141, 2, 3, 5, 7, 12, 13axtgbtwnid 25165 . . . 4 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡)
1514, 11eqeltrd 2688 . . 3 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴𝐷)
16 hpg.o . . . . 5 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
171, 2, 3, 16, 6, 6islnopp 25431 . . . 4 (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐴))))
1817simplbda 652 . . 3 ((𝜑𝐴𝑂𝐴) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐴))
1915, 18r19.29a 3060 . 2 ((𝜑𝐴𝑂𝐴) → 𝐴𝐷)
2017simprbda 651 . . 3 ((𝜑𝐴𝑂𝐴) → (¬ 𝐴𝐷 ∧ ¬ 𝐴𝐷))
2120simpld 474 . 2 ((𝜑𝐴𝑂𝐴) → ¬ 𝐴𝐷)
2219, 21pm2.65da 598 1 (𝜑 → ¬ 𝐴𝑂𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   class class class wbr 4583  {copab 4642  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-trkgb 25148  df-trkg 25152 This theorem is referenced by:  lnoppnhpg  25456
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