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Theorem hpgne2 26475
Description: Points on the open half plane cannot lie on its border. (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)
hpgne1.1 (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)
Assertion
Ref Expression
hpgne2 (𝜑 → ¬ 𝐵𝐷)
Distinct variable groups:   𝑡,𝐴   𝑡,𝐵   𝐷,𝑎,𝑏,𝑡   𝐺,𝑎,𝑏,𝑡   𝐼,𝑎,𝑏,𝑡   𝑡,𝐿   𝑂,𝑎,𝑏,𝑡   𝑃,𝑎,𝑏,𝑡   𝜑,𝑡
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐿(𝑎,𝑏)

Proof of Theorem hpgne2
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 ishpg.p . . 3 𝑃 = (Base‘𝐺)
2 eqid 2818 . . 3 (dist‘𝐺) = (dist‘𝐺)
3 ishpg.i . . 3 𝐼 = (Itv‘𝐺)
4 ishpg.o . . 3 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5 ishpg.l . . 3 𝐿 = (LineG‘𝐺)
6 ishpg.d . . . 4 (𝜑𝐷 ∈ ran 𝐿)
76ad2antrr 722 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿)
8 ishpg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
98ad2antrr 722 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG)
10 hpgbr.b . . . 4 (𝜑𝐵𝑃)
1110ad2antrr 722 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐵𝑃)
12 simplr 765 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝑐𝑃)
13 simprr 769 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐵𝑂𝑐)
141, 2, 3, 4, 5, 7, 9, 11, 12, 13oppne1 26454 . 2 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → ¬ 𝐵𝐷)
15 hpgne1.1 . . 3 (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)
16 hpgbr.a . . . 4 (𝜑𝐴𝑃)
171, 3, 5, 4, 8, 6, 16, 10hpgbr 26473 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
1815, 17mpbid 233 . 2 (𝜑 → ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐))
1914, 18r19.29a 3286 1 (𝜑 → ¬ 𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  cdif 3930   class class class wbr 5057  {copab 5119  ran crn 5549  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149  LineGclng 26150  hpGchpg 26470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-hpg 26471
This theorem is referenced by: (None)
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