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Theorem islnopp 27990
Description: The property for two points 𝐴 and 𝐡 to lie on the opposite sides of a set 𝐷 Definition 9.1 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)
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‘))}
islnopp.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
islnopp.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
Assertion
Ref Expression
islnopp (πœ‘ β†’ (𝐴𝑂𝐡 ↔ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
Distinct variable groups:   𝐷,π‘Ž,𝑏   𝐼,π‘Ž,𝑏   𝑃,π‘Ž,𝑏   𝑑,𝐴   𝑑,𝐡   𝑑,π‘Ž,𝑏
Allowed substitution hints:   πœ‘(𝑑,π‘Ž,𝑏)   𝐴(π‘Ž,𝑏)   𝐡(π‘Ž,𝑏)   𝐷(𝑑)   𝑃(𝑑)   𝐺(𝑑,π‘Ž,𝑏)   𝐼(𝑑)   βˆ’ (𝑑,π‘Ž,𝑏)   𝑂(𝑑,π‘Ž,𝑏)
Proof of Theorem islnopp
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islnopp.a . . 3 (πœ‘ β†’ 𝐴 ∈ 𝑃)
2 islnopp.b . . 3 (πœ‘ β†’ 𝐡 ∈ 𝑃)
3 eleq1 2822 . . . . . 6 (𝑒 = 𝐴 β†’ (𝑒 ∈ (𝑃 βˆ– 𝐷) ↔ 𝐴 ∈ (𝑃 βˆ– 𝐷)))
43anbi1d 631 . . . . 5 (𝑒 = 𝐴 β†’ ((𝑒 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷)) ↔ (𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷))))
5 oveq1 7416 . . . . . . 7 (𝑒 = 𝐴 β†’ (𝑒𝐼𝑣) = (𝐴𝐼𝑣))
65eleq2d 2820 . . . . . 6 (𝑒 = 𝐴 β†’ (𝑑 ∈ (𝑒𝐼𝑣) ↔ 𝑑 ∈ (𝐴𝐼𝑣)))
76rexbidv 3179 . . . . 5 (𝑒 = 𝐴 β†’ (βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝑒𝐼𝑣) ↔ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝑣)))
84, 7anbi12d 632 . . . 4 (𝑒 = 𝐴 β†’ (((𝑒 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝑒𝐼𝑣)) ↔ ((𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝑣))))
9 eleq1 2822 . . . . . 6 (𝑣 = 𝐡 β†’ (𝑣 ∈ (𝑃 βˆ– 𝐷) ↔ 𝐡 ∈ (𝑃 βˆ– 𝐷)))
109anbi2d 630 . . . . 5 (𝑣 = 𝐡 β†’ ((𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷)) ↔ (𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝐡 ∈ (𝑃 βˆ– 𝐷))))
11 oveq2 7417 . . . . . . 7 (𝑣 = 𝐡 β†’ (𝐴𝐼𝑣) = (𝐴𝐼𝐡))
1211eleq2d 2820 . . . . . 6 (𝑣 = 𝐡 β†’ (𝑑 ∈ (𝐴𝐼𝑣) ↔ 𝑑 ∈ (𝐴𝐼𝐡)))
1312rexbidv 3179 . . . . 5 (𝑣 = 𝐡 β†’ (βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝑣) ↔ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡)))
1410, 13anbi12d 632 . . . 4 (𝑣 = 𝐡 β†’ (((𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝑣)) ↔ ((𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝐡 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
15 hpg.o . . . . 5 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
16 simpl 484 . . . . . . . . 9 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ π‘Ž = 𝑒)
1716eleq1d 2819 . . . . . . . 8 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ (π‘Ž ∈ (𝑃 βˆ– 𝐷) ↔ 𝑒 ∈ (𝑃 βˆ– 𝐷)))
18 simpr 486 . . . . . . . . 9 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ 𝑏 = 𝑣)
1918eleq1d 2819 . . . . . . . 8 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ (𝑏 ∈ (𝑃 βˆ– 𝐷) ↔ 𝑣 ∈ (𝑃 βˆ– 𝐷)))
2017, 19anbi12d 632 . . . . . . 7 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ↔ (𝑒 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷))))
21 oveq12 7418 . . . . . . . . 9 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ (π‘ŽπΌπ‘) = (𝑒𝐼𝑣))
2221eleq2d 2820 . . . . . . . 8 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ (𝑑 ∈ (π‘ŽπΌπ‘) ↔ 𝑑 ∈ (𝑒𝐼𝑣)))
2322rexbidv 3179 . . . . . . 7 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ (βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘) ↔ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝑒𝐼𝑣)))
2420, 23anbi12d 632 . . . . . 6 ((π‘Ž = 𝑒 ∧ 𝑏 = 𝑣) β†’ (((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘)) ↔ ((𝑒 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝑒𝐼𝑣))))
2524cbvopabv 5222 . . . . 5 {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))} = {βŸ¨π‘’, π‘£βŸ© ∣ ((𝑒 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝑒𝐼𝑣))}
2615, 25eqtri 2761 . . . 4 𝑂 = {βŸ¨π‘’, π‘£βŸ© ∣ ((𝑒 ∈ (𝑃 βˆ– 𝐷) ∧ 𝑣 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝑒𝐼𝑣))}
278, 14, 26brabg 5540 . . 3 ((𝐴 ∈ 𝑃 ∧ 𝐡 ∈ 𝑃) β†’ (𝐴𝑂𝐡 ↔ ((𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝐡 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
281, 2, 27syl2anc 585 . 2 (πœ‘ β†’ (𝐴𝑂𝐡 ↔ ((𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝐡 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
291biantrurd 534 . . . . 5 (πœ‘ β†’ (Β¬ 𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ Β¬ 𝐴 ∈ 𝐷)))
30 eldif 3959 . . . . 5 (𝐴 ∈ (𝑃 βˆ– 𝐷) ↔ (𝐴 ∈ 𝑃 ∧ Β¬ 𝐴 ∈ 𝐷))
3129, 30bitr4di 289 . . . 4 (πœ‘ β†’ (Β¬ 𝐴 ∈ 𝐷 ↔ 𝐴 ∈ (𝑃 βˆ– 𝐷)))
322biantrurd 534 . . . . 5 (πœ‘ β†’ (Β¬ 𝐡 ∈ 𝐷 ↔ (𝐡 ∈ 𝑃 ∧ Β¬ 𝐡 ∈ 𝐷)))
33 eldif 3959 . . . . 5 (𝐡 ∈ (𝑃 βˆ– 𝐷) ↔ (𝐡 ∈ 𝑃 ∧ Β¬ 𝐡 ∈ 𝐷))
3432, 33bitr4di 289 . . . 4 (πœ‘ β†’ (Β¬ 𝐡 ∈ 𝐷 ↔ 𝐡 ∈ (𝑃 βˆ– 𝐷)))
3531, 34anbi12d 632 . . 3 (πœ‘ β†’ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ↔ (𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝐡 ∈ (𝑃 βˆ– 𝐷))))
3635anbi1d 631 . 2 (πœ‘ β†’ (((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡)) ↔ ((𝐴 ∈ (𝑃 βˆ– 𝐷) ∧ 𝐡 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
3728, 36bitr4d 282 1 (πœ‘ β†’ (𝐴𝑂𝐡 ↔ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ 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-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:  islnoppd  27991  oppne1  27992  oppne2  27993  oppne3  27994  oppcom  27995  oppnid  27997  opphllem1  27998  opphllem3  28000  opphllem5  28002  opphllem6  28003  oppperpex  28004  outpasch  28006  lnopp2hpgb  28014  hpgerlem  28016  colopp  28020  colhp  28021  trgcopyeulem  28056
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