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Theorem hpgne2 27801
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 = (Baseβ€˜πΊ)
ishpg.i 𝐼 = (Itvβ€˜πΊ)
ishpg.l 𝐿 = (LineGβ€˜πΊ)
ishpg.o 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
ishpg.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
ishpg.d (πœ‘ β†’ 𝐷 ∈ ran 𝐿)
hpgbr.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
hpgbr.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
hpgne1.1 (πœ‘ β†’ 𝐴((hpGβ€˜πΊ)β€˜π·)𝐡)
Assertion
Ref Expression
hpgne2 (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐷)
Distinct variable groups:   𝑑,𝐴   𝑑,𝐡   𝐷,π‘Ž,𝑏,𝑑   𝐺,π‘Ž,𝑏,𝑑   𝐼,π‘Ž,𝑏,𝑑   𝑑,𝐿   𝑂,π‘Ž,𝑏,𝑑   𝑃,π‘Ž,𝑏,𝑑   πœ‘,𝑑
Allowed substitution hints:   πœ‘(π‘Ž,𝑏)   𝐴(π‘Ž,𝑏)   𝐡(π‘Ž,𝑏)   𝐿(π‘Ž,𝑏)

Proof of Theorem hpgne2
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 ishpg.p . . 3 𝑃 = (Baseβ€˜πΊ)
2 eqid 2731 . . 3 (distβ€˜πΊ) = (distβ€˜πΊ)
3 ishpg.i . . 3 𝐼 = (Itvβ€˜πΊ)
4 ishpg.o . . 3 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
5 ishpg.l . . 3 𝐿 = (LineGβ€˜πΊ)
6 ishpg.d . . . 4 (πœ‘ β†’ 𝐷 ∈ ran 𝐿)
76ad2antrr 724 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝐷 ∈ ran 𝐿)
8 ishpg.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
98ad2antrr 724 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝐺 ∈ TarskiG)
10 hpgbr.b . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑃)
1110ad2antrr 724 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝐡 ∈ 𝑃)
12 simplr 767 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝑐 ∈ 𝑃)
13 simprr 771 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝐡𝑂𝑐)
141, 2, 3, 4, 5, 7, 9, 11, 12, 13oppne1 27780 . 2 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ Β¬ 𝐡 ∈ 𝐷)
15 hpgne1.1 . . 3 (πœ‘ β†’ 𝐴((hpGβ€˜πΊ)β€˜π·)𝐡)
16 hpgbr.a . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑃)
171, 3, 5, 4, 8, 6, 16, 10hpgbr 27799 . . 3 (πœ‘ β†’ (𝐴((hpGβ€˜πΊ)β€˜π·)𝐡 ↔ βˆƒπ‘ ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)))
1815, 17mpbid 231 . 2 (πœ‘ β†’ βˆƒπ‘ ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐))
1914, 18r19.29a 3161 1 (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3069   βˆ– cdif 3925   class class class wbr 5125  {copab 5187  ran crn 5654  β€˜cfv 6516  (class class class)co 7377  Basecbs 17109  distcds 17171  TarskiGcstrkg 27466  Itvcitv 27472  LineGclng 27473  hpGchpg 27796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-hpg 27797
This theorem is referenced by: (None)
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