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Theorem hpgne1 26539
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 2819 . . 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 724 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿)
8 ishpg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
98ad2antrr 724 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG)
10 hpgbr.a . . . 4 (𝜑𝐴𝑃)
1110ad2antrr 724 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐴𝑃)
12 simplr 767 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝑐𝑃)
13 simprl 769 . . 3 (((𝜑𝑐𝑃) ∧ (𝐴𝑂𝑐𝐵𝑂𝑐)) → 𝐴𝑂𝑐)
141, 2, 3, 4, 5, 7, 9, 11, 12, 13oppne1 26519 . 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 26538 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
1815, 17mpbid 234 . 2 (𝜑 → ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐))
1914, 18r19.29a 3287 1 (𝜑 → ¬ 𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1531  wcel 2108  wrex 3137  cdif 3931   class class class wbr 5057  {copab 5119  ran crn 5549  cfv 6348  (class class class)co 7148  Basecbs 16475  distcds 16566  TarskiGcstrkg 26208  Itvcitv 26214  LineGclng 26215  hpGchpg 26535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 7151  df-hpg 26536
This theorem is referenced by:  colhp  26548  trgcopyeulem  26583
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