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Theorem islbs4 26714
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 5555 . . 3  |-  ( X  e.  (LBasis `  W
)  ->  W  e.  _V )
2 islbs4.j . . 3  |-  J  =  (LBasis `  W )
31, 2eleq2s 2375 . 2  |-  ( X  e.  J  ->  W  e.  _V )
4 elfvex 5555 . . 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 2283 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2283 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
9 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
10 islbs4.k . . . 4  |-  K  =  ( LSpan `  W )
11 eqid 2283 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
126, 7, 8, 9, 2, 10, 11islbs 15829 . . 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 26695 . . . . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LSpanclspn 15728  LBasisclbs 15827  LIndSclinds 26687
This theorem is referenced by:  lbslinds  26715  islinds3  26716  lmimlbs  26718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-lbs 15828  df-lindf 26688  df-linds 26689
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