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Theorem islindf 18974
Description: Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
islindf.b  |-  B  =  ( Base `  W
)
islindf.v  |-  .x.  =  ( .s `  W )
islindf.k  |-  K  =  ( LSpan `  W )
islindf.s  |-  S  =  (Scalar `  W )
islindf.n  |-  N  =  ( Base `  S
)
islindf.z  |-  .0.  =  ( 0g `  S )
Assertion
Ref Expression
islindf  |-  ( ( W  e.  Y  /\  F  e.  X )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  }
)  -.  ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
Distinct variable groups:    k, F, x    k, N    k, W, x    .0. , k
Allowed substitution hints:    B( x, k)    S( x, k)    .x. ( x, k)    K( x, k)    N( x)    X( x, k)    Y( x, k)    .0. ( x)

Proof of Theorem islindf
Dummy variables  f  w  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 5719 . . . . . 6  |-  ( f  =  F  ->  (
f : dom  f --> ( Base `  w )  <->  F : dom  f --> (
Base `  w )
) )
21adantr 465 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f : dom  f
--> ( Base `  w
)  <->  F : dom  f --> ( Base `  w )
) )
3 dmeq 5213 . . . . . . 7  |-  ( f  =  F  ->  dom  f  =  dom  F )
43adantr 465 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  dom  f  =  dom  F )
5 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
6 islindf.b . . . . . . . 8  |-  B  =  ( Base `  W
)
75, 6syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  B )
87adantl 466 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  ( Base `  w
)  =  B )
94, 8feq23d 5732 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( F : dom  f
--> ( Base `  w
)  <->  F : dom  F --> B ) )
102, 9bitrd 253 . . . 4  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f : dom  f
--> ( Base `  w
)  <->  F : dom  F --> B ) )
11 fvex 5882 . . . . . 6  |-  (Scalar `  w )  e.  _V
12 fveq2 5872 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  ( Base `  s )  =  (
Base `  (Scalar `  w
) ) )
13 fveq2 5872 . . . . . . . . . 10  |-  ( s  =  (Scalar `  w
)  ->  ( 0g `  s )  =  ( 0g `  (Scalar `  w ) ) )
1413sneqd 4044 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  { ( 0g `  s ) }  =  { ( 0g
`  (Scalar `  w )
) } )
1512, 14difeq12d 3619 . . . . . . . 8  |-  ( s  =  (Scalar `  w
)  ->  ( ( Base `  s )  \  { ( 0g `  s ) } )  =  ( ( Base `  (Scalar `  w )
)  \  { ( 0g `  (Scalar `  w
) ) } ) )
1615raleqdv 3060 . . . . . . 7  |-  ( s  =  (Scalar `  w
)  ->  ( A. k  e.  ( ( Base `  s )  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w ) ( f `  x
) )  e.  ( ( LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )  <->  A. k  e.  (
( Base `  (Scalar `  w
) )  \  {
( 0g `  (Scalar `  w ) ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) ) ) )
1716ralbidv 2896 . . . . . 6  |-  ( s  =  (Scalar `  w
)  ->  ( A. x  e.  dom  f A. k  e.  ( ( Base `  s )  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w ) ( f `  x
) )  e.  ( ( LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )  <->  A. x  e.  dom  f A. k  e.  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  -.  (
k ( .s `  w ) ( f `
 x ) )  e.  ( ( LSpan `  w ) `  (
f " ( dom  f  \  { x } ) ) ) ) )
1811, 17sbcie 3362 . . . . 5  |-  ( [. (Scalar `  w )  / 
s ]. A. x  e. 
dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  <->  A. x  e.  dom  f A. k  e.  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  -.  (
k ( .s `  w ) ( f `
 x ) )  e.  ( ( LSpan `  w ) `  (
f " ( dom  f  \  { x } ) ) ) )
19 fveq2 5872 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
20 islindf.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
2119, 20syl6eqr 2516 . . . . . . . . . . 11  |-  ( w  =  W  ->  (Scalar `  w )  =  S )
2221fveq2d 5876 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  S )
)
23 islindf.n . . . . . . . . . 10  |-  N  =  ( Base `  S
)
2422, 23syl6eqr 2516 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  N )
2521fveq2d 5876 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  S ) )
26 islindf.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  S )
2725, 26syl6eqr 2516 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
2827sneqd 4044 . . . . . . . . 9  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
2924, 28difeq12d 3619 . . . . . . . 8  |-  ( w  =  W  ->  (
( Base `  (Scalar `  w
) )  \  {
( 0g `  (Scalar `  w ) ) } )  =  ( N 
\  {  .0.  }
) )
3029adantl 466 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  =  ( N  \  {  .0.  } ) )
31 fveq2 5872 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
32 islindf.v . . . . . . . . . . . 12  |-  .x.  =  ( .s `  W )
3331, 32syl6eqr 2516 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
3433adantl 466 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( .s `  w
)  =  .x.  )
35 eqidd 2458 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  k  =  k )
36 fveq1 5871 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
3736adantr 465 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f `  x
)  =  ( F `
 x ) )
3834, 35, 37oveq123d 6317 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( k ( .s
`  w ) ( f `  x ) )  =  ( k 
.x.  ( F `  x ) ) )
39 fveq2 5872 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
40 islindf.k . . . . . . . . . . . 12  |-  K  =  ( LSpan `  W )
4139, 40syl6eqr 2516 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSpan `  w )  =  K )
4241adantl 466 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( LSpan `  w )  =  K )
43 imaeq1 5342 . . . . . . . . . . . 12  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  f  \  { x } ) ) )
443difeq1d 3617 . . . . . . . . . . . . 13  |-  ( f  =  F  ->  ( dom  f  \  { x } )  =  ( dom  F  \  {
x } ) )
4544imaeq2d 5347 . . . . . . . . . . . 12  |-  ( f  =  F  ->  ( F " ( dom  f  \  { x } ) )  =  ( F
" ( dom  F  \  { x } ) ) )
4643, 45eqtrd 2498 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  F  \  { x } ) ) )
4746adantr 465 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f " ( dom  f  \  { x } ) )  =  ( F " ( dom  F  \  { x } ) ) )
4842, 47fveq12d 5878 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  =  ( K `  ( F
" ( dom  F  \  { x } ) ) ) )
4938, 48eleq12d 2539 . . . . . . . 8  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( k ( .s `  w ) ( f `  x
) )  e.  ( ( LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )  <-> 
( k  .x.  ( F `  x )
)  e.  ( K `
 ( F "
( dom  F  \  {
x } ) ) ) ) )
5049notbid 294 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  W )  ->  ( -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  <->  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) )
5130, 50raleqbidv 3068 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  ( A. k  e.  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  -.  (
k ( .s `  w ) ( f `
 x ) )  e.  ( ( LSpan `  w ) `  (
f " ( dom  f  \  { x } ) ) )  <->  A. k  e.  ( N  \  {  .0.  }
)  -.  ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) )
524, 51raleqbidv 3068 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( A. x  e. 
dom  f A. k  e.  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  -.  (
k ( .s `  w ) ( f `
 x ) )  e.  ( ( LSpan `  w ) `  (
f " ( dom  f  \  { x } ) ) )  <->  A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  } )  -.  ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) )
5318, 52syl5bb 257 . . . 4  |-  ( ( f  =  F  /\  w  =  W )  ->  ( [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  <->  A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  } )  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) )
5410, 53anbi12d 710 . . 3  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( f : dom  f --> ( Base `  w )  /\  [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) ) )  <->  ( F : dom  F --> B  /\  A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  }
)  -.  ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
55 df-lindf 18968 . . 3  |- LIndF  =  { <. f ,  w >.  |  ( f : dom  f
--> ( Base `  w
)  /\  [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) ) ) }
5654, 55brabga 4770 . 2  |-  ( ( F  e.  X  /\  W  e.  Y )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  }
)  -.  ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
5756ancoms 453 1  |-  ( ( W  e.  Y  /\  F  e.  X )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  }
)  -.  ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   [.wsbc 3327    \ cdif 3468   {csn 4032   class class class wbr 4456   dom cdm 5008   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14644  Scalarcsca 14715   .scvsca 14716   0gc0g 14857   LSpanclspn 17744   LIndF clindf 18966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-lindf 18968
This theorem is referenced by:  islinds2  18975  islindf2  18976  lindff  18977  lindfind  18978  f1lindf  18984  lsslindf  18992
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