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Theorem lsatlspsn2 33807
Description: The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 33808 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 995 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( X  e.  V  /\  X  =/=  .0.  ) )
2 eldifsn 4152 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  <->  ( X  e.  V  /\  X  =/= 
.0.  ) )
31, 2sylibr 212 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
4 eqid 2467 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
5 sneq 4037 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
65fveq2d 5870 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
76eqeq2d 2481 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
87rspcev 3214 . . 3  |-  ( ( X  e.  ( V 
\  {  .0.  }
)  /\  ( N `  { X } )  =  ( N `  { X } ) )  ->  E. v  e.  ( V  \  {  .0.  } ) ( N `  { X } )  =  ( N `  {
v } ) )
93, 4, 8sylancl 662 . 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 33806 . . 3  |-  ( W  e.  LMod  ->  ( ( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
15143ad2ant1 1017 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  (
( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
169, 15mpbird 232 1  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( N `  { X } )  e.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473   {csn 4027   ` cfv 5588   Basecbs 14490   0gc0g 14695   LModclmod 17312   LSpanclspn 17417  LSAtomsclsa 33789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-lsatoms 33791
This theorem is referenced by:  lsatel  33820  lsmsat  33823  lssatomic  33826  lssats  33827  dihlsprn  36146  dihatlat  36149  dihatexv  36153  dochsatshpb  36267
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