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Theorem lsatlspsn2 29182
Description: The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 29183 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 3749 . . . 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 2283 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
5 sneq 3651 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
65fveq2d 5529 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
76eqeq2d 2294 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
87rspcev 2884 . . 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 29181 . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSpanclspn 15728  LSAtomsclsa 29164
This theorem is referenced by:  lsatel  29195  lsmsat  29198  lssatomic  29201  lssats  29202  dihlsprn  31521  dihatlat  31524  dihatexv  31528  dochsatshpb  31642
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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