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

Theorem lsatset 29107
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 2909 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5670 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lsatset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2439 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5670 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
7 lsatset.z . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
86, 7syl6eqr 2439 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
98sneqd 3772 . . . . . . 7  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
105, 9difeq12d 3411 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  \  { ( 0g `  w ) } )  =  ( V  \  {  .0.  } ) )
11 fveq2 5670 . . . . . . . 8  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
12 lsatset.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
1311, 12syl6eqr 2439 . . . . . . 7  |-  ( w  =  W  ->  ( LSpan `  w )  =  N )
1413fveq1d 5672 . . . . . 6  |-  ( w  =  W  ->  (
( LSpan `  w ) `  { v } )  =  ( N `  { v } ) )
1510, 14mpteq12dv 4230 . . . . 5  |-  ( w  =  W  ->  (
v  e.  ( (
Base `  w )  \  { ( 0g `  w ) } ) 
|->  ( ( LSpan `  w
) `  { v } ) )  =  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
1615rneqd 5039 . . . 4  |-  ( w  =  W  ->  ran  ( v  e.  ( ( Base `  w
)  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  {
v } ) )  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
17 df-lsatoms 29093 . . . 4  |- LSAtoms  =  ( w  e.  _V  |->  ran  ( v  e.  ( ( Base `  w
)  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  {
v } ) ) )
18 fvex 5684 . . . . . . . 8  |-  ( LSpan `  W )  e.  _V
1912, 18eqeltri 2459 . . . . . . 7  |-  N  e. 
_V
2019rnex 5075 . . . . . 6  |-  ran  N  e.  _V
21 p0ex 4329 . . . . . 6  |-  { (/) }  e.  _V
2220, 21unex 4649 . . . . 5  |-  ( ran 
N  u.  { (/) } )  e.  _V
23 eqid 2389 . . . . . . 7  |-  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )  =  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )
24 fvrn0 5695 . . . . . . . 8  |-  ( N `
 { v } )  e.  ( ran 
N  u.  { (/) } )
2524a1i 11 . . . . . . 7  |-  ( v  e.  ( V  \  {  .0.  } )  -> 
( N `  {
v } )  e.  ( ran  N  u.  {
(/) } ) )
2623, 25fmpti 5833 . . . . . 6  |-  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) : ( V  \  {  .0.  } ) --> ( ran 
N  u.  { (/) } )
27 frn 5539 . . . . . 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 4291 . . . 4  |-  ran  (
v  e.  ( V 
\  {  .0.  }
)  |->  ( N `  { v } ) )  e.  _V
3016, 17, 29fvmpt 5747 . . 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 2433 1  |-  ( W  e.  X  ->  A  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `
 { v } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2901    \ cdif 3262    u. cun 3263    C_ wss 3265   (/)c0 3573   {csn 3759    e. cmpt 4209   ran crn 4821   -->wf 5392   ` cfv 5396   Basecbs 13398   0gc0g 13652   LSpanclspn 15976  LSAtomsclsa 29091
This theorem is referenced by:  islsat  29108  lsatlss  29113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-lsatoms 29093
  Copyright terms: Public domain W3C validator