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Theorem islnopp 25321
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 2580 . . . . . 6 (𝑢 = 𝐴 → (𝑢 ∈ (𝑃𝐷) ↔ 𝐴 ∈ (𝑃𝐷)))
43anbi1d 736 . . . . 5 (𝑢 = 𝐴 → ((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ↔ (𝐴 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷))))
5 id 22 . . . . . . . 8 (𝑢 = 𝐴𝑢 = 𝐴)
65oveq1d 6441 . . . . . . 7 (𝑢 = 𝐴 → (𝑢𝐼𝑣) = (𝐴𝐼𝑣))
76eleq2d 2577 . . . . . 6 (𝑢 = 𝐴 → (𝑡 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝑣)))
87rexbidv 2938 . . . . 5 (𝑢 = 𝐴 → (∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝑣)))
94, 8anbi12d 742 . . . 4 (𝑢 = 𝐴 → (((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣)) ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝑣))))
10 eleq1 2580 . . . . . 6 (𝑣 = 𝐵 → (𝑣 ∈ (𝑃𝐷) ↔ 𝐵 ∈ (𝑃𝐷)))
1110anbi2d 735 . . . . 5 (𝑣 = 𝐵 → ((𝐴 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ↔ (𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷))))
12 oveq2 6434 . . . . . . 7 (𝑣 = 𝐵 → (𝐴𝐼𝑣) = (𝐴𝐼𝐵))
1312eleq2d 2577 . . . . . 6 (𝑣 = 𝐵 → (𝑡 ∈ (𝐴𝐼𝑣) ↔ 𝑡 ∈ (𝐴𝐼𝐵)))
1413rexbidv 2938 . . . . 5 (𝑣 = 𝐵 → (∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝑣) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
1511, 14anbi12d 742 . . . 4 (𝑣 = 𝐵 → (((𝐴 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝑣)) ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
16 hpg.o . . . . 5 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
17 simpl 471 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
18 eqidd 2515 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑃𝐷) = (𝑃𝐷))
1917, 18eleq12d 2586 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎 ∈ (𝑃𝐷) ↔ 𝑢 ∈ (𝑃𝐷)))
20 simpr 475 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
2120, 18eleq12d 2586 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏 ∈ (𝑃𝐷) ↔ 𝑣 ∈ (𝑃𝐷)))
2219, 21anbi12d 742 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ↔ (𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷))))
23 oveq12 6435 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
2423eleq2d 2577 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑢𝐼𝑣)))
2524rexbidv 2938 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣)))
2622, 25anbi12d 742 . . . . . 6 ((𝑎 = 𝑢𝑏 = 𝑣) → (((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣))))
2726cbvopabv 4552 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣))}
2816, 27eqtri 2536 . . . 4 𝑂 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃𝐷) ∧ 𝑣 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑢𝐼𝑣))}
299, 15, 28brabg 4813 . . 3 ((𝐴𝑃𝐵𝑃) → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
301, 2, 29syl2anc 690 . 2 (𝜑 → (𝐴𝑂𝐵 ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
311biantrurd 527 . . . . 5 (𝜑 → (¬ 𝐴𝐷 ↔ (𝐴𝑃 ∧ ¬ 𝐴𝐷)))
32 eldif 3454 . . . . 5 (𝐴 ∈ (𝑃𝐷) ↔ (𝐴𝑃 ∧ ¬ 𝐴𝐷))
3331, 32syl6bbr 276 . . . 4 (𝜑 → (¬ 𝐴𝐷𝐴 ∈ (𝑃𝐷)))
342biantrurd 527 . . . . 5 (𝜑 → (¬ 𝐵𝐷 ↔ (𝐵𝑃 ∧ ¬ 𝐵𝐷)))
35 eldif 3454 . . . . 5 (𝐵 ∈ (𝑃𝐷) ↔ (𝐵𝑃 ∧ ¬ 𝐵𝐷))
3634, 35syl6bbr 276 . . . 4 (𝜑 → (¬ 𝐵𝐷𝐵 ∈ (𝑃𝐷)))
3733, 36anbi12d 742 . . 3 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ↔ (𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷))))
3837anbi1d 736 . 2 (𝜑 → (((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)) ↔ ((𝐴 ∈ (𝑃𝐷) ∧ 𝐵 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
3930, 38bitr4d 269 1 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wrex 2801  cdif 3441   class class class wbr 4481  {copab 4540  cfv 5689  (class class class)co 6426  Basecbs 15579  distcds 15661  Itvcitv 25024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-iota 5653  df-fv 5697  df-ov 6429
This theorem is referenced by:  islnoppd  25322  oppne1  25323  oppne2  25324  oppne3  25325  oppcom  25326  oppnid  25328  opphllem1  25329  opphllem2  25330  opphllem3  25331  opphllem4  25332  opphllem5  25333  opphllem6  25334  oppperpex  25335  outpasch  25337  lnopp2hpgb  25345  hpgerlem  25347  colopp  25351  colhp  25352  lmiopp  25384  trgcopyeulem  25387
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