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Theorem lsatset 32475
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 2976 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5686 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lsatset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2488 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5686 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
7 lsatset.z . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
86, 7syl6eqr 2488 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
98sneqd 3884 . . . . . . 7  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
105, 9difeq12d 3470 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  \  { ( 0g `  w ) } )  =  ( V  \  {  .0.  } ) )
11 fveq2 5686 . . . . . . . 8  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
12 lsatset.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
1311, 12syl6eqr 2488 . . . . . . 7  |-  ( w  =  W  ->  ( LSpan `  w )  =  N )
1413fveq1d 5688 . . . . . 6  |-  ( w  =  W  ->  (
( LSpan `  w ) `  { v } )  =  ( N `  { v } ) )
1510, 14mpteq12dv 4365 . . . . 5  |-  ( w  =  W  ->  (
v  e.  ( (
Base `  w )  \  { ( 0g `  w ) } ) 
|->  ( ( LSpan `  w
) `  { v } ) )  =  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
1615rneqd 5062 . . . 4  |-  ( w  =  W  ->  ran  ( v  e.  ( ( Base `  w
)  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  {
v } ) )  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
17 df-lsatoms 32461 . . . 4  |- LSAtoms  =  ( w  e.  _V  |->  ran  ( v  e.  ( ( Base `  w
)  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  {
v } ) ) )
18 fvex 5696 . . . . . . . 8  |-  ( LSpan `  W )  e.  _V
1912, 18eqeltri 2508 . . . . . . 7  |-  N  e. 
_V
2019rnex 6507 . . . . . 6  |-  ran  N  e.  _V
21 p0ex 4474 . . . . . 6  |-  { (/) }  e.  _V
2220, 21unex 6373 . . . . 5  |-  ( ran 
N  u.  { (/) } )  e.  _V
23 eqid 2438 . . . . . . 7  |-  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )  =  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )
24 fvrn0 5707 . . . . . . . 8  |-  ( N `
 { v } )  e.  ( ran 
N  u.  { (/) } )
2524a1i 11 . . . . . . 7  |-  ( v  e.  ( V  \  {  .0.  } )  -> 
( N `  {
v } )  e.  ( ran  N  u.  {
(/) } ) )
2623, 25fmpti 5861 . . . . . 6  |-  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) : ( V  \  {  .0.  } ) --> ( ran 
N  u.  { (/) } )
27 frn 5560 . . . . . 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 5 . . . . 5  |-  ran  (
v  e.  ( V 
\  {  .0.  }
)  |->  ( N `  { v } ) )  C_  ( ran  N  u.  { (/) } )
2922, 28ssexi 4432 . . . 4  |-  ran  (
v  e.  ( V 
\  {  .0.  }
)  |->  ( N `  { v } ) )  e.  _V
3016, 17, 29fvmpt 5769 . . 3  |-  ( W  e.  _V  ->  (LSAtoms `  W )  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
312, 30syl 16 . 2  |-  ( W  e.  X  ->  (LSAtoms `  W )  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
321, 31syl5eq 2482 1  |-  ( W  e.  X  ->  A  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `
 { v } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2967    \ cdif 3320    u. cun 3321    C_ wss 3323   (/)c0 3632   {csn 3872    e. cmpt 4345   ran crn 4836   -->wf 5409   ` cfv 5413   Basecbs 14166   0gc0g 14370   LSpanclspn 17029  LSAtomsclsa 32459
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-lsatoms 32461
This theorem is referenced by:  islsat  32476  lsatlss  32481
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