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Theorem hpgne2 26700
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 2738 . . 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 726 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿)
8 ishpg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
98ad2antrr 726 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG)
10 hpgbr.b . . . 4 (𝜑𝐵𝑃)
1110ad2antrr 726 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐵𝑃)
12 simplr 769 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝑐𝑃)
13 simprr 773 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐵𝑂𝑐)
141, 2, 3, 4, 5, 7, 9, 11, 12, 13oppne1 26679 . 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 26698 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
1815, 17mpbid 235 . 2 (𝜑 → ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐))
1914, 18r19.29a 3198 1 (𝜑 → ¬ 𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2113  wrex 3054  cdif 3838   class class class wbr 5027  {copab 5089  ran crn 5520  cfv 6333  (class class class)co 7164  Basecbs 16579  distcds 16670  TarskiGcstrkg 26368  Itvcitv 26374  LineGclng 26375  hpGchpg 26695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-hpg 26696
This theorem is referenced by: (None)
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