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Theorem islnoppd 26453
Description: Deduce that 𝐴 and 𝐵 lie on opposite sides of line 𝐿. (Contributed by Thierry Arnoux, 16-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))}
islnoppd.a (𝜑𝐴𝑃)
islnoppd.b (𝜑𝐵𝑃)
islnoppd.c (𝜑𝐶𝐷)
islnoppd.1 (𝜑 → ¬ 𝐴𝐷)
islnoppd.2 (𝜑 → ¬ 𝐵𝐷)
islnoppd.3 (𝜑𝐶 ∈ (𝐴𝐼𝐵))
Assertion
Ref Expression
islnoppd (𝜑𝐴𝑂𝐵)
Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐵   𝑡,𝐶   𝑡,𝐷,𝑎,𝑏   𝑡,𝐼   𝜑,𝑡
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐶(𝑎,𝑏)   𝑃(𝑡)   𝐺(𝑡,𝑎,𝑏)   (𝑡,𝑎,𝑏)   𝑂(𝑡,𝑎,𝑏)

Proof of Theorem islnoppd
StepHypRef Expression
1 islnoppd.1 . . 3 (𝜑 → ¬ 𝐴𝐷)
2 islnoppd.2 . . 3 (𝜑 → ¬ 𝐵𝐷)
3 islnoppd.c . . . 4 (𝜑𝐶𝐷)
4 simpr 485 . . . . 5 ((𝜑𝑡 = 𝐶) → 𝑡 = 𝐶)
54eleq1d 2894 . . . 4 ((𝜑𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝐶 ∈ (𝐴𝐼𝐵)))
6 islnoppd.3 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐵))
73, 5, 6rspcedvd 3623 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
81, 2, 7jca31 515 . 2 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
9 hpg.p . . 3 𝑃 = (Base‘𝐺)
10 hpg.d . . 3 = (dist‘𝐺)
11 hpg.i . . 3 𝐼 = (Itv‘𝐺)
12 hpg.o . . 3 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
13 islnoppd.a . . 3 (𝜑𝐴𝑃)
14 islnoppd.b . . 3 (𝜑𝐵𝑃)
159, 10, 11, 12, 13, 14islnopp 26452 . 2 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
168, 15mpbird 258 1 (𝜑𝐴𝑂𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  cdif 3930   class class class wbr 5057  {copab 5119  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  Itvcitv 26149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  opphllem2  26461  opphllem4  26463  outpasch  26468  lmiopp  26515
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