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Theorem lindsind 27278
Description: A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
lindfind.s  |-  .x.  =  ( .s `  W )
lindfind.n  |-  N  =  ( LSpan `  W )
lindfind.l  |-  L  =  (Scalar `  W )
lindfind.z  |-  .0.  =  ( 0g `  L )
lindfind.k  |-  K  =  ( Base `  L
)
Assertion
Ref Expression
lindsind  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( F 
\  { E }
) ) )

Proof of Theorem lindsind
Dummy variables  a 
e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 733 . 2  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  E  e.  F
)
2 eldifsn 3929 . . . 4  |-  ( A  e.  ( K  \  {  .0.  } )  <->  ( A  e.  K  /\  A  =/= 
.0.  ) )
32biimpri 199 . . 3  |-  ( ( A  e.  K  /\  A  =/=  .0.  )  ->  A  e.  ( K  \  {  .0.  } ) )
43adantl 454 . 2  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  A  e.  ( K  \  {  .0.  } ) )
5 elfvdm 5760 . . . . . 6  |-  ( F  e.  (LIndS `  W
)  ->  W  e.  dom LIndS )
6 eqid 2438 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
7 lindfind.s . . . . . . 7  |-  .x.  =  ( .s `  W )
8 lindfind.n . . . . . . 7  |-  N  =  ( LSpan `  W )
9 lindfind.l . . . . . . 7  |-  L  =  (Scalar `  W )
10 lindfind.k . . . . . . 7  |-  K  =  ( Base `  L
)
11 lindfind.z . . . . . . 7  |-  .0.  =  ( 0g `  L )
126, 7, 8, 9, 10, 11islinds2 27274 . . . . . 6  |-  ( W  e.  dom LIndS  ->  ( F  e.  (LIndS `  W
)  <->  ( F  C_  ( Base `  W )  /\  A. e  e.  F  A. a  e.  ( K  \  {  .0.  }
)  -.  ( a 
.x.  e )  e.  ( N `  ( F  \  { e } ) ) ) ) )
135, 12syl 16 . . . . 5  |-  ( F  e.  (LIndS `  W
)  ->  ( F  e.  (LIndS `  W )  <->  ( F  C_  ( Base `  W )  /\  A. e  e.  F  A. a  e.  ( K  \  {  .0.  } )  -.  ( a  .x.  e )  e.  ( N `  ( F 
\  { e } ) ) ) ) )
1413ibi 234 . . . 4  |-  ( F  e.  (LIndS `  W
)  ->  ( F  C_  ( Base `  W
)  /\  A. e  e.  F  A. a  e.  ( K  \  {  .0.  } )  -.  (
a  .x.  e )  e.  ( N `  ( F  \  { e } ) ) ) )
1514simprd 451 . . 3  |-  ( F  e.  (LIndS `  W
)  ->  A. e  e.  F  A. a  e.  ( K  \  {  .0.  } )  -.  (
a  .x.  e )  e.  ( N `  ( F  \  { e } ) ) )
1615ad2antrr 708 . 2  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  A. e  e.  F  A. a  e.  ( K  \  {  .0.  }
)  -.  ( a 
.x.  e )  e.  ( N `  ( F  \  { e } ) ) )
17 oveq2 6092 . . . . 5  |-  ( e  =  E  ->  (
a  .x.  e )  =  ( a  .x.  E ) )
18 sneq 3827 . . . . . . 7  |-  ( e  =  E  ->  { e }  =  { E } )
1918difeq2d 3467 . . . . . 6  |-  ( e  =  E  ->  ( F  \  { e } )  =  ( F 
\  { E }
) )
2019fveq2d 5735 . . . . 5  |-  ( e  =  E  ->  ( N `  ( F  \  { e } ) )  =  ( N `
 ( F  \  { E } ) ) )
2117, 20eleq12d 2506 . . . 4  |-  ( e  =  E  ->  (
( a  .x.  e
)  e.  ( N `
 ( F  \  { e } ) )  <->  ( a  .x.  E )  e.  ( N `  ( F 
\  { E }
) ) ) )
2221notbid 287 . . 3  |-  ( e  =  E  ->  ( -.  ( a  .x.  e
)  e.  ( N `
 ( F  \  { e } ) )  <->  -.  ( a  .x.  E )  e.  ( N `  ( F 
\  { E }
) ) ) )
23 oveq1 6091 . . . . 5  |-  ( a  =  A  ->  (
a  .x.  E )  =  ( A  .x.  E ) )
2423eleq1d 2504 . . . 4  |-  ( a  =  A  ->  (
( a  .x.  E
)  e.  ( N `
 ( F  \  { E } ) )  <-> 
( A  .x.  E
)  e.  ( N `
 ( F  \  { E } ) ) ) )
2524notbid 287 . . 3  |-  ( a  =  A  ->  ( -.  ( a  .x.  E
)  e.  ( N `
 ( F  \  { E } ) )  <->  -.  ( A  .x.  E
)  e.  ( N `
 ( F  \  { E } ) ) ) )
2622, 25rspc2va 3061 . 2  |-  ( ( ( E  e.  F  /\  A  e.  ( K  \  {  .0.  }
) )  /\  A. e  e.  F  A. a  e.  ( K  \  {  .0.  } )  -.  ( a  .x.  e )  e.  ( N `  ( F 
\  { e } ) ) )  ->  -.  ( A  .x.  E
)  e.  ( N `
 ( F  \  { E } ) ) )
271, 4, 16, 26syl21anc 1184 1  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( F 
\  { E }
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    \ cdif 3319    C_ wss 3322   {csn 3816   dom cdm 4881   ` cfv 5457  (class class class)co 6084   Basecbs 13474  Scalarcsca 13537   .scvsca 13538   0gc0g 13728   LSpanclspn 16052  LIndSclinds 27266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-lindf 27267  df-linds 27268
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