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Theorem hpgbr 28783
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 28782 . . . 4 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
8 simpl 482 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
98breq1d 5158 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎𝑂𝑐𝑢𝑂𝑐))
10 simpr 484 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
1110breq1d 5158 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏𝑂𝑐𝑣𝑂𝑐))
129, 11anbi12d 632 . . . . . 6 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎𝑂𝑐𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐𝑣𝑂𝑐)))
1312rexbidv 3177 . . . . 5 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐) ↔ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)))
1413cbvopabv 5221 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
157, 14eqtrdi 2791 . . 3 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)})
1615breqd 5159 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵))
17 hpgbr.a . . 3 (𝜑𝐴𝑃)
18 hpgbr.b . . 3 (𝜑𝐵𝑃)
19 simpl 482 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑢 = 𝐴)
2019breq1d 5158 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑢𝑂𝑐𝐴𝑂𝑐))
21 simpr 484 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2221breq1d 5158 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑣𝑂𝑐𝐵𝑂𝑐))
2320, 22anbi12d 632 . . . . 5 ((𝑢 = 𝐴𝑣 = 𝐵) → ((𝑢𝑂𝑐𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐𝐵𝑂𝑐)))
2423rexbidv 3177 . . . 4 ((𝑢 = 𝐴𝑣 = 𝐵) → (∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐) ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
25 eqid 2735 . . . 4 {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
2624, 25brabga 5544 . . 3 ((𝐴𝑃𝐵𝑃) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2717, 18, 26syl2anc 584 . 2 (𝜑 → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2816, 27bitrd 279 1 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  cdif 3960   class class class wbr 5148  {copab 5210  ran crn 5690  cfv 6563  (class class class)co 7431  Basecbs 17245  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  hpGchpg 28780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-hpg 28781
This theorem is referenced by:  hpgne1  28784  hpgne2  28785  lnopp2hpgb  28786  hpgid  28789  hpgcom  28790  hpgtr  28791
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