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Theorem lsatlspsn2 29108
Description: The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 29109 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lsatset.v  |-  V  =  ( Base `  W
)
lsatset.n  |-  N  =  ( LSpan `  W )
lsatset.z  |-  .0.  =  ( 0g `  W )
lsatset.a  |-  A  =  (LSAtoms `  W )
Assertion
Ref Expression
lsatlspsn2  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( N `  { X } )  e.  A
)

Proof of Theorem lsatlspsn2
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 3simpc 956 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( X  e.  V  /\  X  =/=  .0.  ) )
2 eldifsn 3871 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  <->  ( X  e.  V  /\  X  =/= 
.0.  ) )
31, 2sylibr 204 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
4 eqid 2388 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
5 sneq 3769 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
65fveq2d 5673 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
76eqeq2d 2399 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
87rspcev 2996 . . 3  |-  ( ( X  e.  ( V 
\  {  .0.  }
)  /\  ( N `  { X } )  =  ( N `  { X } ) )  ->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) )
93, 4, 8sylancl 644 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  E. v  e.  ( V  \  {  .0.  } ) ( N `
 { X }
)  =  ( N `
 { v } ) )
10 lsatset.v . . . 4  |-  V  =  ( Base `  W
)
11 lsatset.n . . . 4  |-  N  =  ( LSpan `  W )
12 lsatset.z . . . 4  |-  .0.  =  ( 0g `  W )
13 lsatset.a . . . 4  |-  A  =  (LSAtoms `  W )
1410, 11, 12, 13islsat 29107 . . 3  |-  ( W  e.  LMod  ->  ( ( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
15143ad2ant1 978 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  (
( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
169, 15mpbird 224 1  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( N `  { X } )  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    \ cdif 3261   {csn 3758   ` cfv 5395   Basecbs 13397   0gc0g 13651   LModclmod 15878   LSpanclspn 15975  LSAtomsclsa 29090
This theorem is referenced by:  lsatel  29121  lsmsat  29124  lssatomic  29127  lssats  29128  dihlsprn  31447  dihatlat  31450  dihatexv  31454  dochsatshpb  31568
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-lsatoms 29092
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