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Theorem hpgbr 25452
Description: Half-planes : property for points 𝐴 and 𝐵 to belong to the same open half plane delimited by line 𝐷. Definition 9.7 of [Schwabhauser] p. 71. (Contributed by Thierry Arnoux, 4-Mar-2020.)
Hypotheses
Ref Expression
ishpg.p 𝑃 = (Base‘𝐺)
ishpg.i 𝐼 = (Itv‘𝐺)
ishpg.l 𝐿 = (LineG‘𝐺)
ishpg.o 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
ishpg.g (𝜑𝐺 ∈ TarskiG)
ishpg.d (𝜑𝐷 ∈ ran 𝐿)
hpgbr.a (𝜑𝐴𝑃)
hpgbr.b (𝜑𝐵𝑃)
Assertion
Ref Expression
hpgbr (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐   𝐷,𝑎,𝑏,𝑐,𝑡   𝐺,𝑎,𝑏   𝐼,𝑎,𝑏,𝑐,𝑡   𝑂,𝑎,𝑏   𝑃,𝑎,𝑏,𝑐,𝑡
Allowed substitution hints:   𝜑(𝑡,𝑎,𝑏,𝑐)   𝐴(𝑡,𝑎,𝑏)   𝐵(𝑡,𝑎,𝑏)   𝐺(𝑡,𝑐)   𝐿(𝑡,𝑎,𝑏,𝑐)   𝑂(𝑡,𝑐)

Proof of Theorem hpgbr
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishpg.p . . . . 5 𝑃 = (Base‘𝐺)
2 ishpg.i . . . . 5 𝐼 = (Itv‘𝐺)
3 ishpg.l . . . . 5 𝐿 = (LineG‘𝐺)
4 ishpg.o . . . . 5 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5 ishpg.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
6 ishpg.d . . . . 5 (𝜑𝐷 ∈ ran 𝐿)
71, 2, 3, 4, 5, 6ishpg 25451 . . . 4 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
8 simpl 472 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
98breq1d 4593 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎𝑂𝑐𝑢𝑂𝑐))
10 simpr 476 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
1110breq1d 4593 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏𝑂𝑐𝑣𝑂𝑐))
129, 11anbi12d 743 . . . . . 6 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎𝑂𝑐𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐𝑣𝑂𝑐)))
1312rexbidv 3034 . . . . 5 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐) ↔ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)))
1413cbvopabv 4654 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
157, 14syl6eq 2660 . . 3 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)})
1615breqd 4594 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵))
17 hpgbr.a . . 3 (𝜑𝐴𝑃)
18 hpgbr.b . . 3 (𝜑𝐵𝑃)
19 simpl 472 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑢 = 𝐴)
2019breq1d 4593 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑢𝑂𝑐𝐴𝑂𝑐))
21 simpr 476 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2221breq1d 4593 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑣𝑂𝑐𝐵𝑂𝑐))
2320, 22anbi12d 743 . . . . 5 ((𝑢 = 𝐴𝑣 = 𝐵) → ((𝑢𝑂𝑐𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐𝐵𝑂𝑐)))
2423rexbidv 3034 . . . 4 ((𝑢 = 𝐴𝑣 = 𝐵) → (∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐) ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
25 eqid 2610 . . . 4 {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
2624, 25brabga 4914 . . 3 ((𝐴𝑃𝐵𝑃) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2717, 18, 26syl2anc 691 . 2 (𝜑 → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2816, 27bitrd 267 1 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wrex 2897  cdif 3537   class class class wbr 4583  {copab 4642  ran crn 5039  cfv 5804  (class class class)co 6549  Basecbs 15695  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136  hpGchpg 25449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-hpg 25450
This theorem is referenced by:  hpgne1  25453  hpgne2  25454  lnopp2hpgb  25455  hpgid  25458  hpgcom  25459  hpgtr  25460
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