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Theorem oppnid 26540
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 729 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐺 ∈ TarskiG)
6 oppnid.1 . . . . . 6 (𝜑𝐴𝑃)
76ad3antrrr 729 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴𝑃)
8 opphl.l . . . . . 6 𝐿 = (LineG‘𝐺)
9 opphl.d . . . . . . 7 (𝜑𝐷 ∈ ran 𝐿)
109ad3antrrr 729 . . . . . 6 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐷 ∈ ran 𝐿)
11 simplr 768 . . . . . 6 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡𝐷)
121, 8, 3, 5, 10, 11tglnpt 26343 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡𝑃)
13 simpr 488 . . . . 5 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴))
141, 2, 3, 5, 7, 12, 13axtgbtwnid 26260 . . . 4 ((((𝜑𝐴𝑂𝐴) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡)
1514, 11eqeltrd 2890 . . 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 26533 . . . 4 (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐴))))
1817simplbda 503 . . 3 ((𝜑𝐴𝑂𝐴) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐴))
1915, 18r19.29a 3248 . 2 ((𝜑𝐴𝑂𝐴) → 𝐴𝐷)
2017simprbda 502 . . 3 ((𝜑𝐴𝑂𝐴) → (¬ 𝐴𝐷 ∧ ¬ 𝐴𝐷))
2120simpld 498 . 2 ((𝜑𝐴𝑂𝐴) → ¬ 𝐴𝐷)
2219, 21pm2.65da 816 1 (𝜑 → ¬ 𝐴𝑂𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2111  wrex 3107  cdif 3878   class class class wbr 5030  {copab 5092  ran crn 5520  cfv 6324  (class class class)co 7135  Basecbs 16475  distcds 16566  TarskiGcstrkg 26224  Itvcitv 26230  LineGclng 26231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-cnv 5527  df-dm 5529  df-rn 5530  df-iota 6283  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-trkgb 26243  df-trkg 26247
This theorem is referenced by:  lnoppnhpg  26558
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