Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islbs4 Unicode version

Theorem islbs4 27405
Description: A basis is an independent spanning set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
islbs4.b  |-  B  =  ( Base `  W
)
islbs4.j  |-  J  =  (LBasis `  W )
islbs4.k  |-  K  =  ( LSpan `  W )
Assertion
Ref Expression
islbs4  |-  ( X  e.  J  <->  ( X  e.  (LIndS `  W )  /\  ( K `  X
)  =  B ) )

Proof of Theorem islbs4
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5571 . . 3  |-  ( X  e.  (LBasis `  W
)  ->  W  e.  _V )
2 islbs4.j . . 3  |-  J  =  (LBasis `  W )
31, 2eleq2s 2388 . 2  |-  ( X  e.  J  ->  W  e.  _V )
4 elfvex 5571 . . 3  |-  ( X  e.  (LIndS `  W
)  ->  W  e.  _V )
54adantr 451 . 2  |-  ( ( X  e.  (LIndS `  W )  /\  ( K `  X )  =  B )  ->  W  e.  _V )
6 islbs4.b . . . 4  |-  B  =  ( Base `  W
)
7 eqid 2296 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2296 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
9 eqid 2296 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
10 islbs4.k . . . 4  |-  K  =  ( LSpan `  W )
11 eqid 2296 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
126, 7, 8, 9, 2, 10, 11islbs 15845 . . 3  |-  ( W  e.  _V  ->  ( X  e.  J  <->  ( X  C_  B  /\  ( K `
 X )  =  B  /\  A. x  e.  X  A. k  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
k ( .s `  W ) x )  e.  ( K `  ( X  \  { x } ) ) ) ) )
136, 8, 10, 7, 9, 11islinds2 27386 . . . . 5  |-  ( W  e.  _V  ->  ( X  e.  (LIndS `  W
)  <->  ( X  C_  B  /\  A. x  e.  X  A. k  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
k ( .s `  W ) x )  e.  ( K `  ( X  \  { x } ) ) ) ) )
1413anbi1d 685 . . . 4  |-  ( W  e.  _V  ->  (
( X  e.  (LIndS `  W )  /\  ( K `  X )  =  B )  <->  ( ( X  C_  B  /\  A. x  e.  X  A. k  e.  ( ( Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } )  -.  ( k ( .s `  W
) x )  e.  ( K `  ( X  \  { x }
) ) )  /\  ( K `  X )  =  B ) ) )
15 3anan32 946 . . . 4  |-  ( ( X  C_  B  /\  ( K `  X )  =  B  /\  A. x  e.  X  A. k  e.  ( ( Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } )  -.  ( k ( .s `  W
) x )  e.  ( K `  ( X  \  { x }
) ) )  <->  ( ( X  C_  B  /\  A. x  e.  X  A. k  e.  ( ( Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } )  -.  ( k ( .s `  W
) x )  e.  ( K `  ( X  \  { x }
) ) )  /\  ( K `  X )  =  B ) )
1614, 15syl6rbbr 255 . . 3  |-  ( W  e.  _V  ->  (
( X  C_  B  /\  ( K `  X
)  =  B  /\  A. x  e.  X  A. k  e.  ( ( Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } )  -.  ( k ( .s `  W
) x )  e.  ( K `  ( X  \  { x }
) ) )  <->  ( X  e.  (LIndS `  W )  /\  ( K `  X
)  =  B ) ) )
1712, 16bitrd 244 . 2  |-  ( W  e.  _V  ->  ( X  e.  J  <->  ( X  e.  (LIndS `  W )  /\  ( K `  X
)  =  B ) ) )
183, 5, 17pm5.21nii 342 1  |-  ( X  e.  J  <->  ( X  e.  (LIndS `  W )  /\  ( K `  X
)  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LSpanclspn 15744  LBasisclbs 15843  LIndSclinds 27378
This theorem is referenced by:  lbslinds  27406  islinds3  27407  lmimlbs  27409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-lbs 15844  df-lindf 27379  df-linds 27380
  Copyright terms: Public domain W3C validator