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Theorem islsat 29863
Description: The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lsatset.v  |-  V  =  ( Base `  W
)
lsatset.n  |-  N  =  ( LSpan `  W )
lsatset.z  |-  .0.  =  ( 0g `  W )
lsatset.a  |-  A  =  (LSAtoms `  W )
Assertion
Ref Expression
islsat  |-  ( W  e.  X  ->  ( U  e.  A  <->  E. x  e.  ( V  \  {  .0.  } ) U  =  ( N `  {
x } ) ) )
Distinct variable groups:    x, W    x, X    x, N    x, U    x, V    x,  .0.
Allowed substitution hint:    A( x)

Proof of Theorem islsat
StepHypRef Expression
1 lsatset.v . . . 4  |-  V  =  ( Base `  W
)
2 lsatset.n . . . 4  |-  N  =  ( LSpan `  W )
3 lsatset.z . . . 4  |-  .0.  =  ( 0g `  W )
4 lsatset.a . . . 4  |-  A  =  (LSAtoms `  W )
51, 2, 3, 4lsatset 29862 . . 3  |-  ( W  e.  X  ->  A  =  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `
 { x }
) ) )
65eleq2d 2505 . 2  |-  ( W  e.  X  ->  ( U  e.  A  <->  U  e.  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) ) ) )
7 eqid 2438 . . 3  |-  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )
8 fvex 5745 . . 3  |-  ( N `
 { x }
)  e.  _V
97, 8elrnmpti 5124 . 2  |-  ( U  e.  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )  <->  E. x  e.  ( V  \  {  .0.  } ) U  =  ( N `  {
x } ) )
106, 9syl6bb 254 1  |-  ( W  e.  X  ->  ( U  e.  A  <->  E. x  e.  ( V  \  {  .0.  } ) U  =  ( N `  {
x } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   E.wrex 2708    \ cdif 3319   {csn 3816    e. cmpt 4269   ran crn 4882   ` cfv 5457   Basecbs 13474   0gc0g 13728   LSpanclspn 16052  LSAtomsclsa 29846
This theorem is referenced by:  lsatlspsn2  29864  lsatlspsn  29865  islsati  29866  lsateln0  29867  lsatn0  29871  lsatcmp  29875  lsmsat  29880  lsatfixedN  29881  islshpat  29889  lsatcv0  29903  lsat0cv  29905  lcv1  29913  l1cvpat  29926  dih1dimatlem  32201  dihlatat  32209  dochsatshp  32323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-lsatoms 29848
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