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Theorem islbs4 26973
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 5700 . . 3  |-  ( X  e.  (LBasis `  W
)  ->  W  e.  _V )
2 islbs4.j . . 3  |-  J  =  (LBasis `  W )
31, 2eleq2s 2481 . 2  |-  ( X  e.  J  ->  W  e.  _V )
4 elfvex 5700 . . 3  |-  ( X  e.  (LIndS `  W
)  ->  W  e.  _V )
54adantr 452 . 2  |-  ( ( X  e.  (LIndS `  W )  /\  ( K `  X )  =  B )  ->  W  e.  _V )
6 islbs4.b . . . 4  |-  B  =  ( Base `  W
)
7 eqid 2389 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2389 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
9 eqid 2389 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
10 islbs4.k . . . 4  |-  K  =  ( LSpan `  W )
11 eqid 2389 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
126, 7, 8, 9, 2, 10, 11islbs 16077 . . 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 26954 . . . . 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 686 . . . 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 948 . . . 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 256 . . 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 245 . 2  |-  ( W  e.  _V  ->  ( X  e.  J  <->  ( X  e.  (LIndS `  W )  /\  ( K `  X
)  =  B ) ) )
183, 5, 17pm5.21nii 343 1  |-  ( X  e.  J  <->  ( X  e.  (LIndS `  W )  /\  ( K `  X
)  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   _Vcvv 2901    \ cdif 3262    C_ wss 3265   {csn 3759   ` cfv 5396  (class class class)co 6022   Basecbs 13398  Scalarcsca 13461   .scvsca 13462   0gc0g 13652   LSpanclspn 15976  LBasisclbs 16075  LIndSclinds 26946
This theorem is referenced by:  lbslinds  26974  islinds3  26975  lmimlbs  26977
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-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-lbs 16076  df-lindf 26947  df-linds 26948
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