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Theorem hpgne2 26071
 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 2825 . . 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 719 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿)
8 ishpg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
98ad2antrr 719 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG)
10 hpgbr.b . . . 4 (𝜑𝐵𝑃)
1110ad2antrr 719 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐵𝑃)
12 simplr 787 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝑐𝑃)
13 simprr 791 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐵𝑂𝑐)
141, 2, 3, 4, 5, 7, 9, 11, 12, 13oppne1 26050 . 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 26069 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
1815, 17mpbid 224 . 2 (𝜑 → ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐))
1914, 18r19.29a 3288 1 (𝜑 → ¬ 𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∃wrex 3118   ∖ cdif 3795   class class class wbr 4873  {copab 4935  ran crn 5343  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  distcds 16314  TarskiGcstrkg 25742  Itvcitv 25748  LineGclng 25749  hpGchpg 26066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-hpg 26067 This theorem is referenced by: (None)
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