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Theorem islnoppd 28029
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 485 . . . . 5 ((πœ‘ ∧ 𝑑 = 𝐢) β†’ 𝑑 = 𝐢)
54eleq1d 2818 . . . 4 ((πœ‘ ∧ 𝑑 = 𝐢) β†’ (𝑑 ∈ (𝐴𝐼𝐡) ↔ 𝐢 ∈ (𝐴𝐼𝐡)))
6 islnoppd.3 . . . 4 (πœ‘ β†’ 𝐢 ∈ (𝐴𝐼𝐡))
73, 5, 6rspcedvd 3614 . . 3 (πœ‘ β†’ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))
81, 2, 7jca31 515 . 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 28028 . 2 (πœ‘ β†’ (𝐴𝑂𝐡 ↔ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
168, 15mpbird 256 1 (πœ‘ β†’ 𝐴𝑂𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βˆ– cdif 3945   class class class wbr 5148  {copab 5210  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  distcds 17208  Itvcitv 27722
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7414
This theorem is referenced by:  opphllem2  28037  opphllem4  28039  outpasch  28044  lmiopp  28091
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