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

Theorem islnoppd 28763
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 484 . . . . 5 ((𝜑𝑡 = 𝐶) → 𝑡 = 𝐶)
54eleq1d 2824 . . . 4 ((𝜑𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝐶 ∈ (𝐴𝐼𝐵)))
6 islnoppd.3 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐵))
73, 5, 6rspcedvd 3624 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
81, 2, 7jca31 514 . 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 28762 . 2 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
168, 15mpbird 257 1 (𝜑𝐴𝑂𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  cdif 3960   class class class wbr 5148  {copab 5210  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  Itvcitv 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  opphllem2  28771  opphllem4  28773  outpasch  28778  lmiopp  28825
  Copyright terms: Public domain W3C validator