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Theorem islindf 18246
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 5547 . . . . . 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 5045 . . . . . . 7  |-  ( f  =  F  ->  dom  f  =  dom  F )
43adantr 465 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  dom  f  =  dom  F )
5 fveq2 5696 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
6 islindf.b . . . . . . . 8  |-  B  =  ( Base `  W
)
75, 6syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  B )
87adantl 466 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  ( Base `  w
)  =  B )
94, 8feq23d 5559 . . . . 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 5706 . . . . . 6  |-  (Scalar `  w )  e.  _V
12 fveq2 5696 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  ( Base `  s )  =  (
Base `  (Scalar `  w
) ) )
13 fveq2 5696 . . . . . . . . . 10  |-  ( s  =  (Scalar `  w
)  ->  ( 0g `  s )  =  ( 0g `  (Scalar `  w ) ) )
1413sneqd 3894 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  { ( 0g `  s ) }  =  { ( 0g
`  (Scalar `  w )
) } )
1512, 14difeq12d 3480 . . . . . . . 8  |-  ( s  =  (Scalar `  w
)  ->  ( ( Base `  s )  \  { ( 0g `  s ) } )  =  ( ( Base `  (Scalar `  w )
)  \  { ( 0g `  (Scalar `  w
) ) } ) )
1615raleqdv 2928 . . . . . . 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 2740 . . . . . 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 3226 . . . . 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 5696 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
20 islindf.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
2119, 20syl6eqr 2493 . . . . . . . . . . 11  |-  ( w  =  W  ->  (Scalar `  w )  =  S )
2221fveq2d 5700 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  S )
)
23 islindf.n . . . . . . . . . 10  |-  N  =  ( Base `  S
)
2422, 23syl6eqr 2493 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  N )
2521fveq2d 5700 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  S ) )
26 islindf.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  S )
2725, 26syl6eqr 2493 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
2827sneqd 3894 . . . . . . . . 9  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
2924, 28difeq12d 3480 . . . . . . . 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 5696 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
32 islindf.v . . . . . . . . . . . 12  |-  .x.  =  ( .s `  W )
3331, 32syl6eqr 2493 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
3433adantl 466 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( .s `  w
)  =  .x.  )
35 eqidd 2444 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  k  =  k )
36 fveq1 5695 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
3736adantr 465 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f `  x
)  =  ( F `
 x ) )
3834, 35, 37oveq123d 6117 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( k ( .s
`  w ) ( f `  x ) )  =  ( k 
.x.  ( F `  x ) ) )
39 fveq2 5696 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
40 islindf.k . . . . . . . . . . . 12  |-  K  =  ( LSpan `  W )
4139, 40syl6eqr 2493 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSpan `  w )  =  K )
4241adantl 466 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( LSpan `  w )  =  K )
43 imaeq1 5169 . . . . . . . . . . . 12  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  f  \  { x } ) ) )
443difeq1d 3478 . . . . . . . . . . . . 13  |-  ( f  =  F  ->  ( dom  f  \  { x } )  =  ( dom  F  \  {
x } ) )
4544imaeq2d 5174 . . . . . . . . . . . 12  |-  ( f  =  F  ->  ( F " ( dom  f  \  { x } ) )  =  ( F
" ( dom  F  \  { x } ) ) )
4643, 45eqtrd 2475 . . . . . . . . . . 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 5702 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  =  ( K `  ( F
" ( dom  F  \  { x } ) ) ) )
4938, 48eleq12d 2511 . . . . . . . 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 2936 . . . . . 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 2936 . . . . 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 18240 . . 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 4608 . 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 1369    e. wcel 1756   A.wral 2720   [.wsbc 3191    \ cdif 3330   {csn 3882   class class class wbr 4297   dom cdm 4845   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096   Basecbs 14179  Scalarcsca 14246   .scvsca 14247   0gc0g 14383   LSpanclspn 17057   LIndF clindf 18238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-lindf 18240
This theorem is referenced by:  islinds2  18247  islindf2  18248  lindff  18249  lindfind  18250  f1lindf  18256  lsslindf  18264
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