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Theorem lsatlspsn2 29804
Description: The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 29805 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 954 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( X  e.  V  /\  X  =/=  .0.  ) )
2 eldifsn 3762 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  <->  ( X  e.  V  /\  X  =/= 
.0.  ) )
31, 2sylibr 203 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
4 eqid 2296 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
5 sneq 3664 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
65fveq2d 5545 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
76eqeq2d 2307 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
87rspcev 2897 . . 3  |-  ( ( X  e.  ( V 
\  {  .0.  }
)  /\  ( N `  { X } )  =  ( N `  { X } ) )  ->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) )
93, 4, 8sylancl 643 . 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 29803 . . 3  |-  ( W  e.  LMod  ->  ( ( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
15143ad2ant1 976 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  (
( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
169, 15mpbird 223 1  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( N `  { X } )  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162   {csn 3653   ` cfv 5271   Basecbs 13164   0gc0g 13416   LModclmod 15643   LSpanclspn 15744  LSAtomsclsa 29786
This theorem is referenced by:  lsatel  29817  lsmsat  29820  lssatomic  29823  lssats  29824  dihlsprn  32143  dihatlat  32146  dihatexv  32150  dochsatshpb  32264
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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