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

Theorem lsatset 29715
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 2956 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5720 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lsatset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2485 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5720 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
7 lsatset.z . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
86, 7syl6eqr 2485 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
98sneqd 3819 . . . . . . 7  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
105, 9difeq12d 3458 . . . . . 6  |-  ( w  =  W  ->  (
( Base `  w )  \  { ( 0g `  w ) } )  =  ( V  \  {  .0.  } ) )
11 fveq2 5720 . . . . . . . 8  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
12 lsatset.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
1311, 12syl6eqr 2485 . . . . . . 7  |-  ( w  =  W  ->  ( LSpan `  w )  =  N )
1413fveq1d 5722 . . . . . 6  |-  ( w  =  W  ->  (
( LSpan `  w ) `  { v } )  =  ( N `  { v } ) )
1510, 14mpteq12dv 4279 . . . . 5  |-  ( w  =  W  ->  (
v  e.  ( (
Base `  w )  \  { ( 0g `  w ) } ) 
|->  ( ( LSpan `  w
) `  { v } ) )  =  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
1615rneqd 5089 . . . 4  |-  ( w  =  W  ->  ran  ( v  e.  ( ( Base `  w
)  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  {
v } ) )  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
17 df-lsatoms 29701 . . . 4  |- LSAtoms  =  ( w  e.  _V  |->  ran  ( v  e.  ( ( Base `  w
)  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  {
v } ) ) )
18 fvex 5734 . . . . . . . 8  |-  ( LSpan `  W )  e.  _V
1912, 18eqeltri 2505 . . . . . . 7  |-  N  e. 
_V
2019rnex 5125 . . . . . 6  |-  ran  N  e.  _V
21 p0ex 4378 . . . . . 6  |-  { (/) }  e.  _V
2220, 21unex 4699 . . . . 5  |-  ( ran 
N  u.  { (/) } )  e.  _V
23 eqid 2435 . . . . . . 7  |-  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )  =  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) )
24 fvrn0 5745 . . . . . . . 8  |-  ( N `
 { v } )  e.  ( ran 
N  u.  { (/) } )
2524a1i 11 . . . . . . 7  |-  ( v  e.  ( V  \  {  .0.  } )  -> 
( N `  {
v } )  e.  ( ran  N  u.  {
(/) } ) )
2623, 25fmpti 5884 . . . . . 6  |-  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) : ( V  \  {  .0.  } ) --> ( ran 
N  u.  { (/) } )
27 frn 5589 . . . . . 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 4340 . . . 4  |-  ran  (
v  e.  ( V 
\  {  .0.  }
)  |->  ( N `  { v } ) )  e.  _V
3016, 17, 29fvmpt 5798 . . 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 2479 1  |-  ( W  e.  X  ->  A  =  ran  ( v  e.  ( V  \  {  .0.  } )  |->  ( N `
 { v } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806    e. cmpt 4258   ran crn 4871   -->wf 5442   ` cfv 5446   Basecbs 13461   0gc0g 13715   LSpanclspn 16039  LSAtomsclsa 29699
This theorem is referenced by:  islsat  29716  lsatlss  29721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-lsatoms 29701
  Copyright terms: Public domain W3C validator