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Theorem hpgne1 28279
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
hpgne1 (πœ‘ β†’ Β¬ 𝐴 ∈ 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 hpgne1
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 ishpg.p . . 3 𝑃 = (Baseβ€˜πΊ)
2 eqid 2730 . . 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 722 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝐷 ∈ ran 𝐿)
8 ishpg.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
98ad2antrr 722 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝐺 ∈ TarskiG)
10 hpgbr.a . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑃)
1110ad2antrr 722 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝐴 ∈ 𝑃)
12 simplr 765 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝑐 ∈ 𝑃)
13 simprl 767 . . 3 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ 𝐴𝑂𝑐)
141, 2, 3, 4, 5, 7, 9, 11, 12, 13oppne1 28259 . 2 (((πœ‘ ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)) β†’ Β¬ 𝐴 ∈ 𝐷)
15 hpgne1.1 . . 3 (πœ‘ β†’ 𝐴((hpGβ€˜πΊ)β€˜π·)𝐡)
16 hpgbr.b . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑃)
171, 3, 5, 4, 8, 6, 10, 16hpgbr 28278 . . 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 3160 1 (πœ‘ β†’ Β¬ 𝐴 ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068   βˆ– cdif 3944   class class class wbr 5147  {copab 5209  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  distcds 17210  TarskiGcstrkg 27945  Itvcitv 27951  LineGclng 27952  hpGchpg 28275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-hpg 28276
This theorem is referenced by:  colhp  28288  trgcopyeulem  28323
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