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Theorem oppnid 28755
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 768 . . . . . 6 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡𝐷)
121, 8, 3, 5, 10, 11tglnpt 28558 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡𝑃)
13 simpr 484 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴))
141, 2, 3, 5, 7, 12, 13axtgbtwnid 28475 . . . 4 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡)
1514, 11eqeltrd 2840 . . 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 28748 . . . 4 (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐴))))
1817simplbda 499 . . 3 ((𝜑𝐴𝑂𝐴) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐴))
1915, 18r19.29a 3161 . 2 ((𝜑𝐴𝑂𝐴) → 𝐴𝐷)
2017simprbda 498 . . 3 ((𝜑𝐴𝑂𝐴) → (¬ 𝐴𝐷 ∧ ¬ 𝐴𝐷))
2120simpld 494 . 2 ((𝜑𝐴𝑂𝐴) → ¬ 𝐴𝐷)
2219, 21pm2.65da 816 1 (𝜑 → ¬ 𝐴𝑂𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wcel 2107  wrex 3069  cdif 3947   class class class wbr 5142  {copab 5204  ran crn 5685  cfv 6560  (class class class)co 7432  Basecbs 17248  distcds 17307  TarskiGcstrkg 28436  Itvcitv 28442  LineGclng 28443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695  df-iota 6513  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-trkgb 28458  df-trkg 28462
This theorem is referenced by:  lnoppnhpg  28773
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