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Theorem lindfind 19018
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 753 . 2  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  E  e.  dom  F )
2 eldifsn 4141 . . . 4  |-  ( A  e.  ( K  \  {  .0.  } )  <->  ( A  e.  K  /\  A  =/= 
.0.  ) )
32biimpri 206 . . 3  |-  ( ( A  e.  K  /\  A  =/=  .0.  )  ->  A  e.  ( K  \  {  .0.  } ) )
43adantl 464 . 2  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  A  e.  ( K  \  {  .0.  } ) )
5 simpll 751 . . . 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 14765 . . . . . 6  |-  ( A  e.  K  ->  W  e.  _V )
98ad2antrl 725 . . . . 5  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  W  e.  _V )
10 rellindf 19010 . . . . . . 7  |-  Rel LIndF
1110brrelexi 5029 . . . . . 6  |-  ( F LIndF 
W  ->  F  e.  _V )
1211ad2antrr 723 . . . . 5  |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/= 
.0.  ) )  ->  F  e.  _V )
13 eqid 2454 . . . . . 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 19014 . . . . 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 659 . . . 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 210 . . 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 461 . 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 5848 . . . . . 6  |-  ( e  =  E  ->  ( F `  e )  =  ( F `  E ) )
2221oveq2d 6286 . . . . 5  |-  ( e  =  E  ->  (
a  .x.  ( F `  e ) )  =  ( a  .x.  ( F `  E )
) )
23 sneq 4026 . . . . . . . 8  |-  ( e  =  E  ->  { e }  =  { E } )
2423difeq2d 3608 . . . . . . 7  |-  ( e  =  E  ->  ( dom  F  \  { e } )  =  ( dom  F  \  { E } ) )
2524imaeq2d 5325 . . . . . 6  |-  ( e  =  E  ->  ( F " ( dom  F  \  { e } ) )  =  ( F
" ( dom  F  \  { E } ) ) )
2625fveq2d 5852 . . . . 5  |-  ( e  =  E  ->  ( N `  ( F " ( dom  F  \  { e } ) ) )  =  ( N `  ( F
" ( dom  F  \  { E } ) ) ) )
2722, 26eleq12d 2536 . . . 4  |-  ( e  =  E  ->  (
( a  .x.  ( F `  e )
)  e.  ( N `
 ( F "
( dom  F  \  {
e } ) ) )  <->  ( a  .x.  ( F `  E ) )  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) ) )
2827notbid 292 . . 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 6277 . . . . 5  |-  ( a  =  A  ->  (
a  .x.  ( F `  E ) )  =  ( A  .x.  ( F `  E )
) )
3029eleq1d 2523 . . . 4  |-  ( a  =  A  ->  (
( a  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) )  <-> 
( A  .x.  ( F `  E )
)  e.  ( N `
 ( F "
( dom  F  \  { E } ) ) ) ) )
3130notbid 292 . . 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 3217 . 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 1225 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    \ cdif 3458   {csn 4016   class class class wbr 4439   dom cdm 4988   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14716  Scalarcsca 14787   .scvsca 14788   0gc0g 14929   LSpanclspn 17812   LIndF clindf 19006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-slot 14720  df-base 14721  df-lindf 19008
This theorem is referenced by:  lindfind2  19020  lindfrn  19023  f1lindf  19024
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