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Theorem hpgbr 29072
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 29071 . . . 4 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
8 simpl 488 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
98breq1d 5124 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎𝑂𝑐𝑢𝑂𝑐))
10 simpr 490 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
1110breq1d 5124 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏𝑂𝑐𝑣𝑂𝑐))
129, 11anbi12d 644 . . . . . 6 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎𝑂𝑐𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐𝑣𝑂𝑐)))
1312rexbidv 3192 . . . . 5 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐) ↔ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)))
1413cbvopabv 5189 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
157, 14eqtrdi 2817 . . 3 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)})
1615breqd 5125 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵))
17 hpgbr.a . . 3 (𝜑𝐴𝑃)
18 hpgbr.b . . 3 (𝜑𝐵𝑃)
19 simpl 488 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑢 = 𝐴)
2019breq1d 5124 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑢𝑂𝑐𝐴𝑂𝑐))
21 simpr 490 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2221breq1d 5124 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑣𝑂𝑐𝐵𝑂𝑐))
2320, 22anbi12d 644 . . . . 5 ((𝑢 = 𝐴𝑣 = 𝐵) → ((𝑢𝑂𝑐𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐𝐵𝑂𝑐)))
2423rexbidv 3192 . . . 4 ((𝑢 = 𝐴𝑣 = 𝐵) → (∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐) ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
25 eqid 2766 . . . 4 {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
2624, 25brabga 5523 . . 3 ((𝐴𝑃𝐵𝑃) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2717, 18, 26syl2anc 596 . 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
This proof depends on syntax axioms:  wi 4  wb 209  wa 401   = wceq 1570  wcel 2146  wrex 3092  cdif 3905   class class class wbr 5114  {copab 5178  ran crn 5667  cfv 6543  (class class class)co 7423  Basecbs 17294  TarskiGcstrkg 28726  Itvcitv 28732  LineGclng 28733  hpGchpg 29069
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-hpg 29070
This theorem is used by:  hpgne1  29073  hpgne2  29074  lnopp2hpgb  29075  hpgid  29078  hpgcom  29079  hpgtr  29080
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