Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  islnoppd Structured version   Visualization version   GIF version

Theorem islnoppd 25432
 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 476 . . . . 5 ((𝜑𝑡 = 𝐶) → 𝑡 = 𝐶)
54eleq1d 2672 . . . 4 ((𝜑𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝐶 ∈ (𝐴𝐼𝐵)))
6 islnoppd.3 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐵))
73, 5, 6rspcedvd 3289 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
81, 2, 7jca31 555 . 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 25431 . 2 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
168, 15mpbird 246 1 (𝜑𝐴𝑂𝐵)
 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  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  Itvcitv 25135 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-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-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  outpasch  25447
 Copyright terms: Public domain W3C validator