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Theorem oppnid 28754
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 730 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐺 ∈ TarskiG)
6 oppnid.1 . . . . . 6 (𝜑𝐴𝑃)
76ad3antrrr 730 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴𝑃)
8 opphl.l . . . . . 6 𝐿 = (LineG‘𝐺)
9 opphl.d . . . . . . 7 (𝜑𝐷 ∈ ran 𝐿)
109ad3antrrr 730 . . . . . 6 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐷 ∈ ran 𝐿)
11 simplr 769 . . . . . 6 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡𝐷)
121, 8, 3, 5, 10, 11tglnpt 28557 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡𝑃)
13 simpr 484 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴))
141, 2, 3, 5, 7, 12, 13axtgbtwnid 28474 . . . 4 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡)
1514, 11eqeltrd 2841 . . 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 28747 . . . 4 (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐴))))
1817simplbda 499 . . 3 ((𝜑𝐴𝑂𝐴) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐴))
1915, 18r19.29a 3162 . 2 ((𝜑𝐴𝑂𝐴) → 𝐴𝐷)
2017simprbda 498 . . 3 ((𝜑𝐴𝑂𝐴) → (¬ 𝐴𝐷 ∧ ¬ 𝐴𝐷))
2120simpld 494 . 2 ((𝜑𝐴𝑂𝐴) → ¬ 𝐴𝐷)
2219, 21pm2.65da 817 1 (𝜑 → ¬ 𝐴𝑂𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  cdif 3948   class class class wbr 5143  {copab 5205  ran crn 5686  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkgb 28457  df-trkg 28461
This theorem is referenced by:  lnoppnhpg  28772
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