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Theorem lsatset 28553
Description: The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
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
lsatset  |-  ( W  e.  X  ->  A  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `
 { v } ) ) )
Distinct variable groups:    v, N    v, V    v, W    v,  .0.    v, X
Allowed substitution hint:    A( v)

Proof of Theorem lsatset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 lsatset.a . 2  |-  A  =  (LSAtoms `  W )
2 elex 2796 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lsatset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5525 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
7 lsatset.z . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
86, 7syl6eqr 2333 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
98sneqd 3653 . . . . . . 7  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
105, 9difeq12d 3295 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  \  { ( 0g `  w ) } )  =  ( V  \  {  .0.  } ) )
11 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
12 lsatset.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
1311, 12syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  ( LSpan `  w )  =  N )
1413fveq1d 5527 . . . . . 6  |-  ( w  =  W  ->  (
( LSpan `  w ) `  { v } )  =  ( N `  { v } ) )
1510, 14mpteq12dv 4098 . . . . 5  |-  ( w  =  W  ->  (
v  e.  ( (
Base `  w )  \  { ( 0g `  w ) } ) 
|->  ( ( LSpan `  w
) `  { v } ) )  =  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
1615rneqd 4906 . . . 4  |-  ( w  =  W  ->  ran  ( v  e.  ( ( Base `  w
)  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  {
v } ) )  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
17 df-lsatoms 28539 . . . 4  |- LSAtoms  =  ( w  e.  _V  |->  ran  ( v  e.  ( ( Base `  w
)  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  {
v } ) ) )
18 fvex 5539 . . . . . . . 8  |-  ( LSpan `  W )  e.  _V
1912, 18eqeltri 2353 . . . . . . 7  |-  N  e. 
_V
2019rnex 4942 . . . . . 6  |-  ran  N  e.  _V
21 snex 4216 . . . . . 6  |-  { (/) }  e.  _V
2220, 21unex 4518 . . . . 5  |-  ( ran 
N  u.  { (/) } )  e.  _V
23 eqid 2283 . . . . . . 7  |-  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )  =  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )
24 fvrn0 5550 . . . . . . . 8  |-  ( N `
 { v } )  e.  ( ran 
N  u.  { (/) } )
2524a1i 10 . . . . . . 7  |-  ( v  e.  ( V  \  {  .0.  } )  -> 
( N `  {
v } )  e.  ( ran  N  u.  {
(/) } ) )
2623, 25fmpti 5683 . . . . . 6  |-  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) : ( V  \  {  .0.  } ) --> ( ran 
N  u.  { (/) } )
27 frn 5395 . . . . . 6  |-  ( ( v  e.  ( V 
\  {  .0.  }
)  |->  ( N `  { v } ) ) : ( V 
\  {  .0.  }
) --> ( ran  N  u.  { (/) } )  ->  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )  C_  ( ran  N  u.  { (/) } ) )
2826, 27ax-mp 8 . . . . 5  |-  ran  (
v  e.  ( V 
\  {  .0.  }
)  |->  ( N `  { v } ) )  C_  ( ran  N  u.  { (/) } )
2922, 28ssexi 4159 . . . 4  |-  ran  (
v  e.  ( V 
\  {  .0.  }
)  |->  ( N `  { v } ) )  e.  _V
3016, 17, 29fvmpt 5602 . . 3  |-  ( W  e.  _V  ->  (LSAtoms `  W )  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
312, 30syl 15 . 2  |-  ( W  e.  X  ->  (LSAtoms `  W )  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
321, 31syl5eq 2327 1  |-  ( W  e.  X  ->  A  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `
 { v } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255   Basecbs 13148   0gc0g 13400   LSpanclspn 15728  LSAtomsclsa 28537
This theorem is referenced by:  islsat  28554  lsatlss  28559
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 28539
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