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Theorem lindfind 27286
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 731 . 2  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  E  e.  dom  F )
2 eldifsn 3749 . . . 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 LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  A  e.  ( K  \  {  .0.  } ) )
5 simpll 730 . . . 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 13191 . . . . . 6  |-  ( A  e.  K  ->  W  e.  _V )
98ad2antrl 708 . . . . 5  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  W  e.  _V )
10 rellindf 27278 . . . . . . 7  |-  Rel LIndF
1110brrelexi 4729 . . . . . 6  |-  ( F LIndF 
W  ->  F  e.  _V )
1211ad2antrr 706 . . . . 5  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  F  e.  _V )
13 eqid 2283 . . . . . 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 27282 . . . . 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 642 . . . 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 201 . . 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 449 . 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 5525 . . . . . 6  |-  ( e  =  E  ->  ( F `  e )  =  ( F `  E ) )
2221oveq2d 5874 . . . . 5  |-  ( e  =  E  ->  (
a  .x.  ( F `  e ) )  =  ( a  .x.  ( F `  E )
) )
23 sneq 3651 . . . . . . . 8  |-  ( e  =  E  ->  { e }  =  { E } )
2423difeq2d 3294 . . . . . . 7  |-  ( e  =  E  ->  ( dom  F  \  { e } )  =  ( dom  F  \  { E } ) )
2524imaeq2d 5012 . . . . . 6  |-  ( e  =  E  ->  ( F " ( dom  F  \  { e } ) )  =  ( F
" ( dom  F  \  { E } ) ) )
2625fveq2d 5529 . . . . 5  |-  ( e  =  E  ->  ( N `  ( F " ( dom  F  \  { e } ) ) )  =  ( N `  ( F
" ( dom  F  \  { E } ) ) ) )
2722, 26eleq12d 2351 . . . 4  |-  ( e  =  E  ->  (
( a  .x.  ( F `  e )
)  e.  ( N `
 ( F "
( dom  F  \  {
e } ) ) )  <->  ( a  .x.  ( F `  E ) )  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) ) )
2827notbid 285 . . 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 5865 . . . . 5  |-  ( a  =  A  ->  (
a  .x.  ( F `  E ) )  =  ( A  .x.  ( F `  E )
) )
3029eleq1d 2349 . . . 4  |-  ( a  =  A  ->  (
( a  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) )  <-> 
( A  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) ) )
3130notbid 285 . . 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 2891 . 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 1181 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149   {csn 3640   class class class wbr 4023   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LSpanclspn 15728   LIndF clindf 27274
This theorem is referenced by:  lindfind2  27288  lindfrn  27291  f1lindf  27292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-slot 13152  df-base 13153  df-lindf 27276
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