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Theorem lsatlspsn 29183
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 2283 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
3 sneq 3651 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
43fveq2d 5529 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
54eqeq2d 2294 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
65rspcev 2884 . . 3  |-  ( ( X  e.  ( V 
\  {  .0.  }
)  /\  ( N `  { X } )  =  ( N `  { X } ) )  ->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) )
71, 2, 6sylancl 643 . 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 29181 . . 3  |-  ( W  e.  LMod  ->  ( ( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
148, 13syl 15 . 2  |-  ( ph  ->  ( ( N `  { X } )  e.  A  <->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) ) )
157, 14mpbird 223 1  |-  ( ph  ->  ( N `  { X } )  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSpanclspn 15728  LSAtomsclsa 29164
This theorem is referenced by:  lsatspn0  29190  dvh4dimlem  31633  dochsnshp  31643  lclkrlem2a  31697  lclkrlem2c  31699  lclkrlem2e  31701  lcfrlem20  31752  mapdrvallem2  31835  mapdpglem20  31881  mapdpglem30a  31885  mapdpglem30b  31886  hdmaprnlem3eN  32051  hdmaprnlem16N  32055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-lsatoms 29166
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