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Theorem lindsind 27159
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 732 . 2  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  E  e.  F
)
2 eldifsn 3891 . . . 4  |-  ( A  e.  ( K  \  {  .0.  } )  <->  ( A  e.  K  /\  A  =/= 
.0.  ) )
32biimpri 198 . . 3  |-  ( ( A  e.  K  /\  A  =/=  .0.  )  ->  A  e.  ( K  \  {  .0.  } ) )
43adantl 453 . 2  |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  A  e.  ( K  \  {  .0.  } ) )
5 elfvdm 5720 . . . . . 6  |-  ( F  e.  (LIndS `  W
)  ->  W  e.  dom LIndS )
6 eqid 2408 . . . . . . 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 27155 . . . . . 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 233 . . . 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 450 . . 3  |-  ( F  e.  (LIndS `  W
)  ->  A. e  e.  F  A. a  e.  ( K  \  {  .0.  } )  -.  (
a  .x.  e )  e.  ( N `  ( F  \  { e } ) ) )
1615ad2antrr 707 . 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 6052 . . . . 5  |-  ( e  =  E  ->  (
a  .x.  e )  =  ( a  .x.  E ) )
18 sneq 3789 . . . . . . 7  |-  ( e  =  E  ->  { e }  =  { E } )
1918difeq2d 3429 . . . . . 6  |-  ( e  =  E  ->  ( F  \  { e } )  =  ( F 
\  { E }
) )
2019fveq2d 5695 . . . . 5  |-  ( e  =  E  ->  ( N `  ( F  \  { e } ) )  =  ( N `
 ( F  \  { E } ) ) )
2117, 20eleq12d 2476 . . . 4  |-  ( e  =  E  ->  (
( a  .x.  e
)  e.  ( N `
 ( F  \  { e } ) )  <->  ( a  .x.  E )  e.  ( N `  ( F 
\  { E }
) ) ) )
2221notbid 286 . . 3  |-  ( e  =  E  ->  ( -.  ( a  .x.  e
)  e.  ( N `
 ( F  \  { e } ) )  <->  -.  ( a  .x.  E )  e.  ( N `  ( F 
\  { E }
) ) ) )
23 oveq1 6051 . . . . 5  |-  ( a  =  A  ->  (
a  .x.  E )  =  ( A  .x.  E ) )
2423eleq1d 2474 . . . 4  |-  ( a  =  A  ->  (
( a  .x.  E
)  e.  ( N `
 ( F  \  { E } ) )  <-> 
( A  .x.  E
)  e.  ( N `
 ( F  \  { E } ) ) ) )
2524notbid 286 . . 3  |-  ( a  =  A  ->  ( -.  ( a  .x.  E
)  e.  ( N `
 ( F  \  { E } ) )  <->  -.  ( A  .x.  E
)  e.  ( N `
 ( F  \  { E } ) ) ) )
2622, 25rspc2va 3023 . 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 1183 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670    \ cdif 3281    C_ wss 3284   {csn 3778   dom cdm 4841   ` cfv 5417  (class class class)co 6044   Basecbs 13428  Scalarcsca 13491   .scvsca 13492   0gc0g 13682   LSpanclspn 16006  LIndSclinds 27147
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-lindf 27148  df-linds 27149
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