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Theorem lindfind 27264
Description: A linearly independent family 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
lindfind  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  -.  ( A  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) )

Proof of Theorem lindfind
Dummy variables  a 
e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 733 . 2  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  E  e.  dom  F )
2 eldifsn 3928 . . . 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 LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  A  e.  ( K  \  {  .0.  } ) )
5 simpll 732 . . . 4  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  F LIndF  W )
6 lindfind.l . . . . . . 7  |-  L  =  (Scalar `  W )
7 lindfind.k . . . . . . 7  |-  K  =  ( Base `  L
)
86, 7elbasfv 13513 . . . . . 6  |-  ( A  e.  K  ->  W  e.  _V )
98ad2antrl 710 . . . . 5  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  W  e.  _V )
10 rellindf 27256 . . . . . . 7  |-  Rel LIndF
1110brrelexi 4919 . . . . . 6  |-  ( F LIndF 
W  ->  F  e.  _V )
1211ad2antrr 708 . . . . 5  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  F  e.  _V )
13 eqid 2437 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
14 lindfind.s . . . . . 6  |-  .x.  =  ( .s `  W )
15 lindfind.n . . . . . 6  |-  N  =  ( LSpan `  W )
16 lindfind.z . . . . . 6  |-  .0.  =  ( 0g `  L )
1713, 14, 15, 6, 7, 16islindf 27260 . . . . 5  |-  ( ( W  e.  _V  /\  F  e.  _V )  ->  ( F LIndF  W  <->  ( F : dom  F --> ( Base `  W )  /\  A. e  e.  dom  F A. a  e.  ( K  \  {  .0.  } )  -.  ( a  .x.  ( F `  e ) )  e.  ( N `
 ( F "
( dom  F  \  {
e } ) ) ) ) ) )
189, 12, 17syl2anc 644 . . . 4  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  -> 
( F LIndF  W  <->  ( F : dom  F --> ( Base `  W )  /\  A. e  e.  dom  F A. a  e.  ( K  \  {  .0.  } )  -.  ( a  .x.  ( F `  e ) )  e.  ( N `
 ( F "
( dom  F  \  {
e } ) ) ) ) ) )
195, 18mpbid 203 . . 3  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  -> 
( F : dom  F --> ( Base `  W
)  /\  A. e  e.  dom  F A. a  e.  ( K  \  {  .0.  } )  -.  (
a  .x.  ( F `  e ) )  e.  ( N `  ( F " ( dom  F  \  { e } ) ) ) ) )
2019simprd 451 . 2  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  A. e  e.  dom  F A. a  e.  ( K  \  {  .0.  } )  -.  ( a 
.x.  ( F `  e ) )  e.  ( N `  ( F " ( dom  F  \  { e } ) ) ) )
21 fveq2 5729 . . . . . 6  |-  ( e  =  E  ->  ( F `  e )  =  ( F `  E ) )
2221oveq2d 6098 . . . . 5  |-  ( e  =  E  ->  (
a  .x.  ( F `  e ) )  =  ( a  .x.  ( F `  E )
) )
23 sneq 3826 . . . . . . . 8  |-  ( e  =  E  ->  { e }  =  { E } )
2423difeq2d 3466 . . . . . . 7  |-  ( e  =  E  ->  ( dom  F  \  { e } )  =  ( dom  F  \  { E } ) )
2524imaeq2d 5204 . . . . . 6  |-  ( e  =  E  ->  ( F " ( dom  F  \  { e } ) )  =  ( F
" ( dom  F  \  { E } ) ) )
2625fveq2d 5733 . . . . 5  |-  ( e  =  E  ->  ( N `  ( F " ( dom  F  \  { e } ) ) )  =  ( N `  ( F
" ( dom  F  \  { E } ) ) ) )
2722, 26eleq12d 2505 . . . 4  |-  ( e  =  E  ->  (
( a  .x.  ( F `  e )
)  e.  ( N `
 ( F "
( dom  F  \  {
e } ) ) )  <->  ( a  .x.  ( F `  E ) )  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) ) )
2827notbid 287 . . 3  |-  ( e  =  E  ->  ( -.  ( a  .x.  ( F `  e )
)  e.  ( N `
 ( F "
( dom  F  \  {
e } ) ) )  <->  -.  ( a  .x.  ( F `  E
) )  e.  ( N `  ( F
" ( dom  F  \  { E } ) ) ) ) )
29 oveq1 6089 . . . . 5  |-  ( a  =  A  ->  (
a  .x.  ( F `  E ) )  =  ( A  .x.  ( F `  E )
) )
3029eleq1d 2503 . . . 4  |-  ( a  =  A  ->  (
( a  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) )  <-> 
( A  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) ) )
3130notbid 287 . . 3  |-  ( a  =  A  ->  ( -.  ( a  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) )  <->  -.  ( A  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) ) )
3228, 31rspc2va 3060 . 2  |-  ( ( ( E  e.  dom  F  /\  A  e.  ( K  \  {  .0.  } ) )  /\  A. e  e.  dom  F A. a  e.  ( K  \  {  .0.  } )  -.  ( a  .x.  ( F `  e ) )  e.  ( N `
 ( F "
( dom  F  \  {
e } ) ) ) )  ->  -.  ( A  .x.  ( F `
 E ) )  e.  ( N `  ( F " ( dom 
F  \  { E } ) ) ) )
331, 4, 20, 32syl21anc 1184 1  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  -.  ( A  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   _Vcvv 2957    \ cdif 3318   {csn 3815   class class class wbr 4213   dom cdm 4879   "cima 4882   -->wf 5451   ` cfv 5455  (class class class)co 6082   Basecbs 13470  Scalarcsca 13533   .scvsca 13534   0gc0g 13724   LSpanclspn 16048   LIndF clindf 27252
This theorem is referenced by:  lindfind2  27266  lindfrn  27269  f1lindf  27270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-slot 13474  df-base 13475  df-lindf 27254
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