Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lshpne Structured version   Visualization version   GIF version

Theorem lshpne 38458
Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lshpne.v 𝑉 = (Baseβ€˜π‘Š)
lshpne.h 𝐻 = (LSHypβ€˜π‘Š)
lshpne.w (πœ‘ β†’ π‘Š ∈ LMod)
lshpne.u (πœ‘ β†’ π‘ˆ ∈ 𝐻)
Assertion
Ref Expression
lshpne (πœ‘ β†’ π‘ˆ β‰  𝑉)

Proof of Theorem lshpne
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 lshpne.u . . 3 (πœ‘ β†’ π‘ˆ ∈ 𝐻)
2 lshpne.w . . . 4 (πœ‘ β†’ π‘Š ∈ LMod)
3 lshpne.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
4 eqid 2727 . . . . 5 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
5 eqid 2727 . . . . 5 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
6 lshpne.h . . . . 5 𝐻 = (LSHypβ€˜π‘Š)
73, 4, 5, 6islshp 38455 . . . 4 (π‘Š ∈ LMod β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ (LSubSpβ€˜π‘Š) ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 ((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
82, 7syl 17 . . 3 (πœ‘ β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ (LSubSpβ€˜π‘Š) ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 ((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
91, 8mpbid 231 . 2 (πœ‘ β†’ (π‘ˆ ∈ (LSubSpβ€˜π‘Š) ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 ((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
109simp2d 1140 1 (πœ‘ β†’ π‘ˆ β‰  𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2936  βˆƒwrex 3066   βˆͺ cun 3945  {csn 4630  β€˜cfv 6551  Basecbs 17185  LModclmod 20748  LSubSpclss 20820  LSpanclspn 20860  LSHypclsh 38451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-lshyp 38453
This theorem is referenced by:  lshpnel  38459  lshpcmp  38464  lkrshp3  38582  lkrshp4  38584  dochshpncl  40861  dochlkr  40862  dochkrshp  40863  dochsatshpb  40929
  Copyright terms: Public domain W3C validator