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Theorem hpgne1 28784
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
hpgne1 (𝜑 → ¬ 𝐴𝐷)
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 hpgne1
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 ishpg.p . . 3 𝑃 = (Base‘𝐺)
2 eqid 2735 . . 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.a . . . 4 (𝜑𝐴𝑃)
1110ad2antrr 726 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐴𝑃)
12 simplr 769 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝑐𝑃)
13 simprl 771 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐴𝑂𝑐)
141, 2, 3, 4, 5, 7, 9, 11, 12, 13oppne1 28764 . 2 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → ¬ 𝐴𝐷)
15 hpgne1.1 . . 3 (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)
16 hpgbr.b . . . 4 (𝜑𝐵𝑃)
171, 3, 5, 4, 8, 6, 10, 16hpgbr 28783 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
1815, 17mpbid 232 . 2 (𝜑 → ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐))
1914, 18r19.29a 3160 1 (𝜑 → ¬ 𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  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  distcds 17307  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:  colhp  28793  trgcopyeulem  28828
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