Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lsatlspsn2 Structured version   Unicode version

Theorem lsatlspsn2 32634
Description: The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 32635 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 987 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( X  e.  V  /\  X  =/=  .0.  ) )
2 eldifsn 3998 . . . 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 2441 . . 3  |-  ( N `
 { X }
)  =  ( N `
 { X }
)
5 sneq 3885 . . . . . 6  |-  ( v  =  X  ->  { v }  =  { X } )
65fveq2d 5693 . . . . 5  |-  ( v  =  X  ->  ( N `  { v } )  =  ( N `  { X } ) )
76eqeq2d 2452 . . . 4  |-  ( v  =  X  ->  (
( N `  { X } )  =  ( N `  { v } )  <->  ( N `  { X } )  =  ( N `  { X } ) ) )
87rspcev 3071 . . 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 32633 . . 3  |-  ( W  e.  LMod  ->  ( ( N `  { X } )  e.  A  <->  E. v  e.  ( V 
\  {  .0.  }
) ( N `  { X } )  =  ( N `  {
v } ) ) )
15143ad2ant1 1009 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2604   E.wrex 2714    \ cdif 3323   {csn 3875   ` cfv 5416   Basecbs 14172   0gc0g 14376   LModclmod 16946   LSpanclspn 17050  LSAtomsclsa 32616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-lsatoms 32618
This theorem is referenced by:  lsatel  32647  lsmsat  32650  lssatomic  32653  lssats  32654  dihlsprn  34973  dihatlat  34976  dihatexv  34980  dochsatshpb  35094
  Copyright terms: Public domain W3C validator