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Theorem lsatlspsn 29108
Description: The span of a non-zero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
Hypotheses
Ref Expression
lsatset.v  |-  V  =  ( Base `  W
)
lsatset.n  |-  N  =  ( LSpan `  W )
lsatset.z  |-  .0.  =  ( 0g `  W )
lsatset.a  |-  A  =  (LSAtoms `  W )
lsatlspsn.w  |-  ( ph  ->  W  e.  LMod )
lsatlspsn.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
Assertion
Ref Expression
lsatlspsn  |-  ( ph  ->  ( N `  { X } )  e.  A
)

Proof of Theorem lsatlspsn
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lsatlspsn.x . . 3  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2 eqid 2387 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
3 sneq 3768 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
43fveq2d 5672 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
54eqeq2d 2398 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
65rspcev 2995 . . 3  |-  ( ( X  e.  ( V 
\  {  .0.  }
)  /\  ( N `  { X } )  =  ( N `  { X } ) )  ->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) )
71, 2, 6sylancl 644 . 2  |-  ( ph  ->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) )
8 lsatlspsn.w . . 3  |-  ( ph  ->  W  e.  LMod )
9 lsatset.v . . . 4  |-  V  =  ( Base `  W
)
10 lsatset.n . . . 4  |-  N  =  ( LSpan `  W )
11 lsatset.z . . . 4  |-  .0.  =  ( 0g `  W )
12 lsatset.a . . . 4  |-  A  =  (LSAtoms `  W )
139, 10, 11, 12islsat 29106 . . 3  |-  ( W  e.  LMod  ->  ( ( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
148, 13syl 16 . 2  |-  ( ph  ->  ( ( N `  { X } )  e.  A  <->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) ) )
157, 14mpbird 224 1  |-  ( ph  ->  ( N `  { X } )  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   E.wrex 2650    \ cdif 3260   {csn 3757   ` cfv 5394   Basecbs 13396   0gc0g 13650   LModclmod 15877   LSpanclspn 15974  LSAtomsclsa 29089
This theorem is referenced by:  lsatspn0  29115  dvh4dimlem  31558  dochsnshp  31568  lclkrlem2a  31622  lclkrlem2c  31624  lclkrlem2e  31626  lcfrlem20  31677  mapdrvallem2  31760  mapdpglem20  31806  mapdpglem30a  31810  mapdpglem30b  31811  hdmaprnlem3eN  31976  hdmaprnlem16N  31980
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-lsatoms 29091
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