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Theorem lsatlspsn 29692
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 2435 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
3 sneq 3817 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
43fveq2d 5724 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
54eqeq2d 2446 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
65rspcev 3044 . . 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 29690 . . 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 1652    e. wcel 1725   E.wrex 2698    \ cdif 3309   {csn 3806   ` cfv 5446   Basecbs 13459   0gc0g 13713   LModclmod 15940   LSpanclspn 16037  LSAtomsclsa 29673
This theorem is referenced by:  lsatspn0  29699  dvh4dimlem  32142  dochsnshp  32152  lclkrlem2a  32206  lclkrlem2c  32208  lclkrlem2e  32210  lcfrlem20  32261  mapdrvallem2  32344  mapdpglem20  32390  mapdpglem30a  32394  mapdpglem30b  32395  hdmaprnlem3eN  32560  hdmaprnlem16N  32564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-lsatoms 29675
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