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Theorem lsatlspsn 29805
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 2296 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
3 sneq 3664 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
43fveq2d 5545 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
54eqeq2d 2307 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
65rspcev 2897 . . 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 29803 . . 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 1632    e. wcel 1696   E.wrex 2557    \ cdif 3162   {csn 3653   ` cfv 5271   Basecbs 13164   0gc0g 13416   LModclmod 15643   LSpanclspn 15744  LSAtomsclsa 29786
This theorem is referenced by:  lsatspn0  29812  dvh4dimlem  32255  dochsnshp  32265  lclkrlem2a  32319  lclkrlem2c  32321  lclkrlem2e  32323  lcfrlem20  32374  mapdrvallem2  32457  mapdpglem20  32503  mapdpglem30a  32507  mapdpglem30b  32508  hdmaprnlem3eN  32673  hdmaprnlem16N  32677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-lsatoms 29788
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