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Theorem hpgbr 26008
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 26007 . . . 4 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
8 simpl 475 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
98breq1d 4853 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎𝑂𝑐𝑢𝑂𝑐))
10 simpr 478 . . . . . . . 8 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
1110breq1d 4853 . . . . . . 7 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏𝑂𝑐𝑣𝑂𝑐))
129, 11anbi12d 625 . . . . . 6 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎𝑂𝑐𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐𝑣𝑂𝑐)))
1312rexbidv 3233 . . . . 5 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐) ↔ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)))
1413cbvopabv 4915 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
157, 14syl6eq 2849 . . 3 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)})
1615breqd 4854 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵))
17 hpgbr.a . . 3 (𝜑𝐴𝑃)
18 hpgbr.b . . 3 (𝜑𝐵𝑃)
19 simpl 475 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑢 = 𝐴)
2019breq1d 4853 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑢𝑂𝑐𝐴𝑂𝑐))
21 simpr 478 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2221breq1d 4853 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐵) → (𝑣𝑂𝑐𝐵𝑂𝑐))
2320, 22anbi12d 625 . . . . 5 ((𝑢 = 𝐴𝑣 = 𝐵) → ((𝑢𝑂𝑐𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐𝐵𝑂𝑐)))
2423rexbidv 3233 . . . 4 ((𝑢 = 𝐴𝑣 = 𝐵) → (∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐) ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
25 eqid 2799 . . . 4 {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}
2624, 25brabga 5185 . . 3 ((𝐴𝑃𝐵𝑃) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2717, 18, 26syl2anc 580 . 2 (𝜑 → (𝐴{⟨𝑢, 𝑣⟩ ∣ ∃𝑐𝑃 (𝑢𝑂𝑐𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
2816, 27bitrd 271 1 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  cdif 3766   class class class wbr 4843  {copab 4905  ran crn 5313  cfv 6101  (class class class)co 6878  Basecbs 16184  TarskiGcstrkg 25681  Itvcitv 25687  LineGclng 25688  hpGchpg 26005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-hpg 26006
This theorem is referenced by:  hpgne1  26009  hpgne2  26010  lnopp2hpgb  26011  hpgid  26014  hpgcom  26015  hpgtr  26016
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