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Theorem islnoppd 26541
 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 488 . . . . 5 ((𝜑𝑡 = 𝐶) → 𝑡 = 𝐶)
54eleq1d 2874 . . . 4 ((𝜑𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝐶 ∈ (𝐴𝐼𝐵)))
6 islnoppd.3 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐵))
73, 5, 6rspcedvd 3574 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
81, 2, 7jca31 518 . 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 26540 . 2 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
168, 15mpbird 260 1 (𝜑𝐴𝑂𝐵)
 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  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  distcds 16568  Itvcitv 26237 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-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-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-iota 6283  df-fv 6332  df-ov 7138 This theorem is referenced by:  opphllem2  26549  opphllem4  26551  outpasch  26556  lmiopp  26603
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