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Theorem hpgbr 26563
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 26562 . . . 4 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
8 simpl 486 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
98breq1d 5063 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎𝑂𝑐𝑢𝑂𝑐))
10 simpr 488 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
1110breq1d 5063 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏𝑂𝑐𝑣𝑂𝑐))
129, 11anbi12d 633 . . . . . 6 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎𝑂𝑐𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐𝑣𝑂𝑐)))
1312rexbidv 3289 . . . . 5 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐) ↔ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)))
1413cbvopabv 5125 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
157, 14syl6eq 2875 . . 3 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)})
1615breqd 5064 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵))
17 hpgbr.a . . 3 (𝜑𝐴𝑃)
18 hpgbr.b . . 3 (𝜑𝐵𝑃)
19 simpl 486 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑢 = 𝐴)
2019breq1d 5063 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑢𝑂𝑐𝐴𝑂𝑐))
21 simpr 488 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2221breq1d 5063 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑣𝑂𝑐𝐵𝑂𝑐))
2320, 22anbi12d 633 . . . . 5 ((𝑢 = 𝐴𝑣 = 𝐵) → ((𝑢𝑂𝑐𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐𝐵𝑂𝑐)))
2423rexbidv 3289 . . . 4 ((𝑢 = 𝐴𝑣 = 𝐵) → (∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐) ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
25 eqid 2824 . . . 4 {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
2624, 25brabga 5409 . . 3 ((𝐴𝑃𝐵𝑃) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2717, 18, 26syl2anc 587 . 2 (𝜑 → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2816, 27bitrd 282 1 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wrex 3134  cdif 3916   class class class wbr 5053  {copab 5115  ran crn 5544  cfv 6345  (class class class)co 7151  Basecbs 16485  TarskiGcstrkg 26233  Itvcitv 26239  LineGclng 26240  hpGchpg 26560
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-hpg 26561
This theorem is referenced by:  hpgne1  26564  hpgne2  26565  lnopp2hpgb  26566  hpgid  26569  hpgcom  26570  hpgtr  26571
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