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

Theorem islsat 29726
Description: The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (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
islsat  |-  ( W  e.  X  ->  ( U  e.  A  <->  E. x  e.  ( V  \  {  .0.  } ) U  =  ( N `  {
x } ) ) )
Distinct variable groups:    x, W    x, X    x, N    x, U    x, V    x,  .0.
Allowed substitution hint:    A( x)

Proof of Theorem islsat
StepHypRef Expression
1 lsatset.v . . . 4  |-  V  =  ( Base `  W
)
2 lsatset.n . . . 4  |-  N  =  ( LSpan `  W )
3 lsatset.z . . . 4  |-  .0.  =  ( 0g `  W )
4 lsatset.a . . . 4  |-  A  =  (LSAtoms `  W )
51, 2, 3, 4lsatset 29725 . . 3  |-  ( W  e.  X  ->  A  =  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `
 { x }
) ) )
65eleq2d 2502 . 2  |-  ( W  e.  X  ->  ( U  e.  A  <->  U  e.  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) ) ) )
7 eqid 2435 . . 3  |-  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )
8 fvex 5734 . . 3  |-  ( N `
 { x }
)  e.  _V
97, 8elrnmpti 5113 . 2  |-  ( U  e.  ran  ( x  e.  ( V  \  {  .0.  } )  |->  ( N `  { x } ) )  <->  E. x  e.  ( V  \  {  .0.  } ) U  =  ( N `  {
x } ) )
106, 9syl6bb 253 1  |-  ( W  e.  X  ->  ( U  e.  A  <->  E. x  e.  ( V  \  {  .0.  } ) U  =  ( N `  {
x } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   E.wrex 2698    \ cdif 3309   {csn 3806    e. cmpt 4258   ran crn 4871   ` cfv 5446   Basecbs 13461   0gc0g 13715   LSpanclspn 16039  LSAtomsclsa 29709
This theorem is referenced by:  lsatlspsn2  29727  lsatlspsn  29728  islsati  29729  lsateln0  29730  lsatn0  29734  lsatcmp  29738  lsmsat  29743  lsatfixedN  29744  islshpat  29752  lsatcv0  29766  lsat0cv  29768  lcv1  29776  l1cvpat  29789  dih1dimatlem  32064  dihlatat  32072  dochsatshp  32186
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 29711
  Copyright terms: Public domain W3C validator