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Theorem lsatlspsn 34861
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 2457 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
3 sneq 4042 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
43fveq2d 5876 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
54eqeq2d 2471 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
65rspcev 3210 . . 3  |-  ( ( X  e.  ( V 
\  {  .0.  }
)  /\  ( N `  { X } )  =  ( N `  { X } ) )  ->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) )
71, 2, 6sylancl 662 . 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 34859 . . 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 232 1  |-  ( ph  ->  ( N `  { X } )  e.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   E.wrex 2808    \ cdif 3468   {csn 4032   ` cfv 5594   Basecbs 14644   0gc0g 14857   LModclmod 17639   LSpanclspn 17744  LSAtomsclsa 34842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-lsatoms 34844
This theorem is referenced by:  lsatspn0  34868  dvh4dimlem  37313  dochsnshp  37323  lclkrlem2a  37377  lclkrlem2c  37379  lclkrlem2e  37381  lcfrlem20  37432  mapdrvallem2  37515  mapdpglem20  37561  mapdpglem30a  37565  mapdpglem30b  37566  hdmaprnlem3eN  37731  hdmaprnlem16N  37735
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