MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oppne1 Structured version   Visualization version   GIF version

Theorem oppne1 28023
Description: Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.)
Hypotheses
Ref Expression
hpg.p 𝑃 = (Baseβ€˜πΊ)
hpg.d βˆ’ = (distβ€˜πΊ)
hpg.i 𝐼 = (Itvβ€˜πΊ)
hpg.o 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
opphl.l 𝐿 = (LineGβ€˜πΊ)
opphl.d (πœ‘ β†’ 𝐷 ∈ ran 𝐿)
opphl.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
oppcom.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
oppcom.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
oppcom.o (πœ‘ β†’ 𝐴𝑂𝐡)
Assertion
Ref Expression
oppne1 (πœ‘ β†’ Β¬ 𝐴 ∈ 𝐷)
Distinct variable groups:   𝐷,π‘Ž,𝑏   𝐼,π‘Ž,𝑏   𝑃,π‘Ž,𝑏   𝑑,𝐴   𝑑,𝐡   𝑑,𝐷   𝑑,𝐺   𝑑,𝐿   𝑑,𝐼   𝑑,𝑂   𝑑,𝑃   πœ‘,𝑑   𝑑, βˆ’   𝑑,π‘Ž,𝑏
Allowed substitution hints:   πœ‘(π‘Ž,𝑏)   𝐴(π‘Ž,𝑏)   𝐡(π‘Ž,𝑏)   𝐺(π‘Ž,𝑏)   𝐿(π‘Ž,𝑏)   βˆ’ (π‘Ž,𝑏)   𝑂(π‘Ž,𝑏)

Proof of Theorem oppne1
StepHypRef Expression
1 oppcom.o . . 3 (πœ‘ β†’ 𝐴𝑂𝐡)
2 hpg.p . . . 4 𝑃 = (Baseβ€˜πΊ)
3 hpg.d . . . 4 βˆ’ = (distβ€˜πΊ)
4 hpg.i . . . 4 𝐼 = (Itvβ€˜πΊ)
5 hpg.o . . . 4 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
6 oppcom.a . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑃)
7 oppcom.b . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑃)
82, 3, 4, 5, 6, 7islnopp 28021 . . 3 (πœ‘ β†’ (𝐴𝑂𝐡 ↔ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
91, 8mpbid 231 . 2 (πœ‘ β†’ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡)))
109simplld 767 1 (πœ‘ β†’ Β¬ 𝐴 ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   βˆ– cdif 3946   class class class wbr 5149  {copab 5211  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  TarskiGcstrkg 27709  Itvcitv 27715  LineGclng 27716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552  df-ov 7412
This theorem is referenced by:  oppne3  28025  opptgdim2  28027  opphllem1  28029  opphllem2  28030  opphllem4  28032  opphl  28036  hpgne1  28043  hpgne2  28044  lnopp2hpgb  28045
  Copyright terms: Public domain W3C validator