MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lssne0 Structured version   Unicode version

Theorem lssne0 17009
Description: A nonzero subspace has a nonzero vector. (shne0i 24802 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
lss0cl.z  |-  .0.  =  ( 0g `  W )
lss0cl.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lssne0  |-  ( X  e.  S  ->  ( X  =/=  {  .0.  }  <->  E. y  e.  X  y  =/=  .0.  ) )
Distinct variable groups:    y, X    y,  .0.
Allowed substitution hints:    S( y)    W( y)

Proof of Theorem lssne0
StepHypRef Expression
1 lss0cl.s . . . . 5  |-  S  =  ( LSubSp `  W )
21lssn0 16999 . . . 4  |-  ( X  e.  S  ->  X  =/=  (/) )
3 eqsn 4029 . . . 4  |-  ( X  =/=  (/)  ->  ( X  =  {  .0.  }  <->  A. y  e.  X  y  =  .0.  ) )
42, 3syl 16 . . 3  |-  ( X  e.  S  ->  ( X  =  {  .0.  }  <->  A. y  e.  X  y  =  .0.  )
)
5 nne 2607 . . . . 5  |-  ( -.  y  =/=  .0.  <->  y  =  .0.  )
65ralbii 2734 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  A. y  e.  X  y  =  .0.  )
7 ralnex 2720 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  -.  E. y  e.  X  y  =/=  .0.  )
86, 7bitr3i 251 . . 3  |-  ( A. y  e.  X  y  =  .0.  <->  -.  E. y  e.  X  y  =/=  .0.  )
94, 8syl6rbb 262 . 2  |-  ( X  e.  S  ->  ( -.  E. y  e.  X  y  =/=  .0.  <->  X  =  {  .0.  } ) )
109necon1abid 2659 1  |-  ( X  e.  S  ->  ( X  =/=  {  .0.  }  <->  E. y  e.  X  y  =/=  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711   (/)c0 3632   {csn 3872   ` cfv 5413   0gc0g 14370   LSubSpclss 16990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-ov 6089  df-lss 16991
This theorem is referenced by:  lsmsat  32493  lssatomic  32496  dochsatshpb  34937  hgmapvvlem3  35413
  Copyright terms: Public domain W3C validator