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Theorem islnopp 28747
Description: The property for two points 𝐴 and 𝐵 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 = (Base‘𝐺)
hpg.d = (dist‘𝐺)
hpg.i 𝐼 = (Itv‘𝐺)
hpg.o 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
islnopp.a (𝜑𝐴𝑃)
islnopp.b (𝜑𝐵𝑃)
Assertion
Ref Expression
islnopp (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐵   𝑡,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑡,𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐷(𝑡)   𝑃(𝑡)   𝐺(𝑡,𝑎,𝑏)   𝐼(𝑡)   (𝑡,𝑎,𝑏)   𝑂(𝑡,𝑎,𝑏)

Proof of Theorem islnopp
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islnopp.a . . 3 (𝜑𝐴𝑃)
2 islnopp.b . . 3 (𝜑𝐵𝑃)
3 eleq1 2829 . . . . . 6 (𝑢 = 𝐴 → (𝑢 ∈ (𝑃𝐷) ↔ 𝐴 ∈ (𝑃𝐷)))
43anbi1d 631 . . . . 5 (𝑢 = 𝐴 → ((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ↔ (𝐴 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷))))
5 oveq1 7438 . . . . . . 7 (𝑢 = 𝐴 → (𝑢𝐼𝑣) = (𝐴𝐼𝑣))
65eleq2d 2827 . . . . . 6 (𝑢 = 𝐴 → (𝑡 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝑣)))
76rexbidv 3179 . . . . 5 (𝑢 = 𝐴 → (∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝑣)))
84, 7anbi12d 632 . . . 4 (𝑢 = 𝐴 → (((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣)) ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝑣))))
9 eleq1 2829 . . . . . 6 (𝑣 = 𝐵 → (𝑣 ∈ (𝑃𝐷) ↔ 𝐵 ∈ (𝑃𝐷)))
109anbi2d 630 . . . . 5 (𝑣 = 𝐵 → ((𝐴 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ↔ (𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷))))
11 oveq2 7439 . . . . . . 7 (𝑣 = 𝐵 → (𝐴𝐼𝑣) = (𝐴𝐼𝐵))
1211eleq2d 2827 . . . . . 6 (𝑣 = 𝐵 → (𝑡 ∈ (𝐴𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝐵)))
1312rexbidv 3179 . . . . 5 (𝑣 = 𝐵 → (∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝑣) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
1410, 13anbi12d 632 . . . 4 (𝑣 = 𝐵 → (((𝐴 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝑣)) ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
15 hpg.o . . . . 5 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
16 simpl 482 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
1716eleq1d 2826 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎 ∈ (𝑃𝐷) ↔ 𝑢 ∈ (𝑃𝐷)))
18 simpr 484 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
1918eleq1d 2826 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏 ∈ (𝑃𝐷) ↔ 𝑣 ∈ (𝑃𝐷)))
2017, 19anbi12d 632 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ↔ (𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷))))
21 oveq12 7440 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
2221eleq2d 2827 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑢𝐼𝑣)))
2322rexbidv 3179 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣)))
2420, 23anbi12d 632 . . . . . 6 ((𝑎 = 𝑢𝑏 = 𝑣) → (((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣))))
2524cbvopabv 5216 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣))}
2615, 25eqtri 2765 . . . 4 𝑂 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣))}
278, 14, 26brabg 5544 . . 3 ((𝐴𝑃𝐵𝑃) → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
281, 2, 27syl2anc 584 . 2 (𝜑 → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
291biantrurd 532 . . . . 5 (𝜑 → (¬ 𝐴𝐷 ↔ (𝐴𝑃 ∧ ¬ 𝐴𝐷)))
30 eldif 3961 . . . . 5 (𝐴 ∈ (𝑃𝐷) ↔ (𝐴𝑃 ∧ ¬ 𝐴𝐷))
3129, 30bitr4di 289 . . . 4 (𝜑 → (¬ 𝐴𝐷𝐴 ∈ (𝑃𝐷)))
322biantrurd 532 . . . . 5 (𝜑 → (¬ 𝐵𝐷 ↔ (𝐵𝑃 ∧ ¬ 𝐵𝐷)))
33 eldif 3961 . . . . 5 (𝐵 ∈ (𝑃𝐷) ↔ (𝐵𝑃 ∧ ¬ 𝐵𝐷))
3432, 33bitr4di 289 . . . 4 (𝜑 → (¬ 𝐵𝐷𝐵 ∈ (𝑃𝐷)))
3531, 34anbi12d 632 . . 3 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ↔ (𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷))))
3635anbi1d 631 . 2 (𝜑 → (((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)) ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
3728, 36bitr4d 282 1 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  cdif 3948   class class class wbr 5143  {copab 5205  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  Itvcitv 28441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  islnoppd  28748  oppne1  28749  oppne2  28750  oppne3  28751  oppcom  28752  oppnid  28754  opphllem1  28755  opphllem3  28757  opphllem5  28759  opphllem6  28760  oppperpex  28761  outpasch  28763  lnopp2hpgb  28771  hpgerlem  28773  colopp  28777  colhp  28778  trgcopyeulem  28813
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