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Theorem islnoppd 27991
Description: Deduce that 𝐴 and 𝐡 lie on opposite sides of line 𝐿. (Contributed by Thierry Arnoux, 16-Aug-2020.)
Hypotheses
Ref Expression
hpg.p 𝑃 = (Baseβ€˜πΊ)
hpg.d βˆ’ = (distβ€˜πΊ)
hpg.i 𝐼 = (Itvβ€˜πΊ)
hpg.o 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
islnoppd.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
islnoppd.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
islnoppd.c (πœ‘ β†’ 𝐢 ∈ 𝐷)
islnoppd.1 (πœ‘ β†’ Β¬ 𝐴 ∈ 𝐷)
islnoppd.2 (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐷)
islnoppd.3 (πœ‘ β†’ 𝐢 ∈ (𝐴𝐼𝐡))
Assertion
Ref Expression
islnoppd (πœ‘ β†’ 𝐴𝑂𝐡)
Distinct variable groups:   𝐷,π‘Ž,𝑏   𝐼,π‘Ž,𝑏   𝑃,π‘Ž,𝑏   𝑑,𝐴   𝑑,𝐡   𝑑,𝐢   𝑑,𝐷,π‘Ž,𝑏   𝑑,𝐼   πœ‘,𝑑
Allowed substitution hints:   πœ‘(π‘Ž,𝑏)   𝐴(π‘Ž,𝑏)   𝐡(π‘Ž,𝑏)   𝐢(π‘Ž,𝑏)   𝑃(𝑑)   𝐺(𝑑,π‘Ž,𝑏)   βˆ’ (𝑑,π‘Ž,𝑏)   𝑂(𝑑,π‘Ž,𝑏)

Proof of Theorem islnoppd
StepHypRef Expression
1 islnoppd.1 . . 3 (πœ‘ β†’ Β¬ 𝐴 ∈ 𝐷)
2 islnoppd.2 . . 3 (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐷)
3 islnoppd.c . . . 4 (πœ‘ β†’ 𝐢 ∈ 𝐷)
4 simpr 486 . . . . 5 ((πœ‘ ∧ 𝑑 = 𝐢) β†’ 𝑑 = 𝐢)
54eleq1d 2819 . . . 4 ((πœ‘ ∧ 𝑑 = 𝐢) β†’ (𝑑 ∈ (𝐴𝐼𝐡) ↔ 𝐢 ∈ (𝐴𝐼𝐡)))
6 islnoppd.3 . . . 4 (πœ‘ β†’ 𝐢 ∈ (𝐴𝐼𝐡))
73, 5, 6rspcedvd 3615 . . 3 (πœ‘ β†’ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))
81, 2, 7jca31 516 . 2 (πœ‘ β†’ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡)))
9 hpg.p . . 3 𝑃 = (Baseβ€˜πΊ)
10 hpg.d . . 3 βˆ’ = (distβ€˜πΊ)
11 hpg.i . . 3 𝐼 = (Itvβ€˜πΊ)
12 hpg.o . . 3 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
13 islnoppd.a . . 3 (πœ‘ β†’ 𝐴 ∈ 𝑃)
14 islnoppd.b . . 3 (πœ‘ β†’ 𝐡 ∈ 𝑃)
159, 10, 11, 12, 13, 14islnopp 27990 . 2 (πœ‘ β†’ (𝐴𝑂𝐡 ↔ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
168, 15mpbird 257 1 (πœ‘ β†’ 𝐴𝑂𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   βˆ– cdif 3946   class class class wbr 5149  {copab 5211  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  Itvcitv 27684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552  df-ov 7412
This theorem is referenced by:  opphllem2  27999  opphllem4  28001  outpasch  28006  lmiopp  28053
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