Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lindfind Unicode version

Theorem lindfind 27389
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 3762 . . . 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 13207 . . . . . 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 27381 . . . . . . 7  |-  Rel LIndF
1110brrelexi 4745 . . . . . 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 2296 . . . . . 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 27385 . . . . 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 5541 . . . . . 6  |-  ( e  =  E  ->  ( F `  e )  =  ( F `  E ) )
2221oveq2d 5890 . . . . 5  |-  ( e  =  E  ->  (
a  .x.  ( F `  e ) )  =  ( a  .x.  ( F `  E )
) )
23 sneq 3664 . . . . . . . 8  |-  ( e  =  E  ->  { e }  =  { E } )
2423difeq2d 3307 . . . . . . 7  |-  ( e  =  E  ->  ( dom  F  \  { e } )  =  ( dom  F  \  { E } ) )
2524imaeq2d 5028 . . . . . 6  |-  ( e  =  E  ->  ( F " ( dom  F  \  { e } ) )  =  ( F
" ( dom  F  \  { E } ) ) )
2625fveq2d 5545 . . . . 5  |-  ( e  =  E  ->  ( N `  ( F " ( dom  F  \  { e } ) ) )  =  ( N `  ( F
" ( dom  F  \  { E } ) ) ) )
2722, 26eleq12d 2364 . . . 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 5881 . . . . 5  |-  ( a  =  A  ->  (
a  .x.  ( F `  E ) )  =  ( A  .x.  ( F `  E )
) )
3029eleq1d 2362 . . . 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 2904 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162   {csn 3653   class class class wbr 4039   dom cdm 4705   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LSpanclspn 15744   LIndF clindf 27377
This theorem is referenced by:  lindfind2  27391  lindfrn  27394  f1lindf  27395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-slot 13168  df-base 13169  df-lindf 27379
  Copyright terms: Public domain W3C validator