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Theorem islsat 29157
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 29156 . . 3  |-  ( W  e.  X  ->  A  =  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `
 { x }
) ) )
65eleq2d 2447 . 2  |-  ( W  e.  X  ->  ( U  e.  A  <->  U  e.  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) ) ) )
7 eqid 2380 . . 3  |-  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )
8 fvex 5675 . . 3  |-  ( N `
 { x }
)  e.  _V
97, 8elrnmpti 5054 . 2  |-  ( U  e.  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )  <->  E. x  e.  ( V  \  {  .0.  } ) U  =  ( N `  {
x } ) )
106, 9syl6bb 253 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 177    = wceq 1649    e. wcel 1717   E.wrex 2643    \ cdif 3253   {csn 3750    e. cmpt 4200   ran crn 4812   ` cfv 5387   Basecbs 13389   0gc0g 13643   LSpanclspn 15967  LSAtomsclsa 29140
This theorem is referenced by:  lsatlspsn2  29158  lsatlspsn  29159  islsati  29160  lsateln0  29161  lsatn0  29165  lsatcmp  29169  lsmsat  29174  lsatfixedN  29175  islshpat  29183  lsatcv0  29197  lsat0cv  29199  lcv1  29207  l1cvpat  29220  dih1dimatlem  31495  dihlatat  31503  dochsatshp  31617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-lsatoms 29142
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