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Theorem lindsind 26610
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 731 . 2  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  E  e.  F
)
2 eldifsn 3825 . . . 4  |-  ( A  e.  ( K  \  {  .0.  } )  <->  ( A  e.  K  /\  A  =/= 
.0.  ) )
32biimpri 197 . . 3  |-  ( ( A  e.  K  /\  A  =/=  .0.  )  ->  A  e.  ( K  \  {  .0.  } ) )
43adantl 452 . 2  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  A  e.  ( K  \  {  .0.  } ) )
5 elfvdm 5637 . . . . . 6  |-  ( F  e.  (LIndS `  W
)  ->  W  e.  dom LIndS )
6 eqid 2358 . . . . . . 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 26606 . . . . . 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 15 . . . . 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 232 . . . 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 449 . . 3  |-  ( F  e.  (LIndS `  W
)  ->  A. e  e.  F  A. a  e.  ( K  \  {  .0.  } )  -.  (
a  .x.  e )  e.  ( N `  ( F  \  { e } ) ) )
1615ad2antrr 706 . 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 5953 . . . . 5  |-  ( e  =  E  ->  (
a  .x.  e )  =  ( a  .x.  E ) )
18 sneq 3727 . . . . . . 7  |-  ( e  =  E  ->  { e }  =  { E } )
1918difeq2d 3370 . . . . . 6  |-  ( e  =  E  ->  ( F  \  { e } )  =  ( F 
\  { E }
) )
2019fveq2d 5612 . . . . 5  |-  ( e  =  E  ->  ( N `  ( F  \  { e } ) )  =  ( N `
 ( F  \  { E } ) ) )
2117, 20eleq12d 2426 . . . 4  |-  ( e  =  E  ->  (
( a  .x.  e
)  e.  ( N `
 ( F  \  { e } ) )  <->  ( a  .x.  E )  e.  ( N `  ( F 
\  { E }
) ) ) )
2221notbid 285 . . 3  |-  ( e  =  E  ->  ( -.  ( a  .x.  e
)  e.  ( N `
 ( F  \  { e } ) )  <->  -.  ( a  .x.  E )  e.  ( N `  ( F 
\  { E }
) ) ) )
23 oveq1 5952 . . . . 5  |-  ( a  =  A  ->  (
a  .x.  E )  =  ( A  .x.  E ) )
2423eleq1d 2424 . . . 4  |-  ( a  =  A  ->  (
( a  .x.  E
)  e.  ( N `
 ( F  \  { E } ) )  <-> 
( A  .x.  E
)  e.  ( N `
 ( F  \  { E } ) ) ) )
2524notbid 285 . . 3  |-  ( a  =  A  ->  ( -.  ( a  .x.  E
)  e.  ( N `
 ( F  \  { E } ) )  <->  -.  ( A  .x.  E
)  e.  ( N `
 ( F  \  { E } ) ) ) )
2622, 25rspc2va 2967 . 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 1181 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 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619    \ cdif 3225    C_ wss 3228   {csn 3716   dom cdm 4771   ` cfv 5337  (class class class)co 5945   Basecbs 13245  Scalarcsca 13308   .scvsca 13309   0gc0g 13499   LSpanclspn 15827  LIndSclinds 26598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-lindf 26599  df-linds 26600
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