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Theorem islindf 27261
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 5578 . . . . . 6  |-  ( f  =  F  ->  (
f : dom  f --> ( Base `  w )  <->  F : dom  f --> (
Base `  w )
) )
21adantr 453 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f : dom  f
--> ( Base `  w
)  <->  F : dom  f --> ( Base `  w )
) )
3 dmeq 5072 . . . . . . 7  |-  ( f  =  F  ->  dom  f  =  dom  F )
43adantr 453 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  dom  f  =  dom  F )
5 fveq2 5730 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
6 islindf.b . . . . . . . 8  |-  B  =  ( Base `  W
)
75, 6syl6eqr 2488 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  B )
87adantl 454 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  ( Base `  w
)  =  B )
94, 8feq23d 5590 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( F : dom  f
--> ( Base `  w
)  <->  F : dom  F --> B ) )
102, 9bitrd 246 . . . 4  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f : dom  f
--> ( Base `  w
)  <->  F : dom  F --> B ) )
11 fvex 5744 . . . . . 6  |-  (Scalar `  w )  e.  _V
12 fveq2 5730 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  ( Base `  s )  =  (
Base `  (Scalar `  w
) ) )
13 fveq2 5730 . . . . . . . . . 10  |-  ( s  =  (Scalar `  w
)  ->  ( 0g `  s )  =  ( 0g `  (Scalar `  w ) ) )
1413sneqd 3829 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  { ( 0g `  s ) }  =  { ( 0g
`  (Scalar `  w )
) } )
1512, 14difeq12d 3468 . . . . . . . 8  |-  ( s  =  (Scalar `  w
)  ->  ( ( Base `  s )  \  { ( 0g `  s ) } )  =  ( ( Base `  (Scalar `  w )
)  \  { ( 0g `  (Scalar `  w
) ) } ) )
1615raleqdv 2912 . . . . . . 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 2727 . . . . . 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 3197 . . . . 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 5730 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
20 islindf.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
2119, 20syl6eqr 2488 . . . . . . . . . . 11  |-  ( w  =  W  ->  (Scalar `  w )  =  S )
2221fveq2d 5734 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  S )
)
23 islindf.n . . . . . . . . . 10  |-  N  =  ( Base `  S
)
2422, 23syl6eqr 2488 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  N )
2521fveq2d 5734 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  S ) )
26 islindf.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  S )
2725, 26syl6eqr 2488 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
2827sneqd 3829 . . . . . . . . 9  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
2924, 28difeq12d 3468 . . . . . . . 8  |-  ( w  =  W  ->  (
( Base `  (Scalar `  w
) )  \  {
( 0g `  (Scalar `  w ) ) } )  =  ( N 
\  {  .0.  }
) )
3029adantl 454 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  =  ( N  \  {  .0.  } ) )
31 fveq2 5730 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
32 islindf.v . . . . . . . . . . . 12  |-  .x.  =  ( .s `  W )
3331, 32syl6eqr 2488 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
3433adantl 454 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( .s `  w
)  =  .x.  )
35 eqidd 2439 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  k  =  k )
36 fveq1 5729 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
3736adantr 453 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f `  x
)  =  ( F `
 x ) )
3834, 35, 37oveq123d 6104 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( k ( .s
`  w ) ( f `  x ) )  =  ( k 
.x.  ( F `  x ) ) )
39 fveq2 5730 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
40 islindf.k . . . . . . . . . . . 12  |-  K  =  ( LSpan `  W )
4139, 40syl6eqr 2488 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSpan `  w )  =  K )
4241adantl 454 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( LSpan `  w )  =  K )
43 imaeq1 5200 . . . . . . . . . . . 12  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  f  \  { x } ) ) )
443difeq1d 3466 . . . . . . . . . . . . 13  |-  ( f  =  F  ->  ( dom  f  \  { x } )  =  ( dom  F  \  {
x } ) )
4544imaeq2d 5205 . . . . . . . . . . . 12  |-  ( f  =  F  ->  ( F " ( dom  f  \  { x } ) )  =  ( F
" ( dom  F  \  { x } ) ) )
4643, 45eqtrd 2470 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  F  \  { x } ) ) )
4746adantr 453 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f " ( dom  f  \  { x } ) )  =  ( F " ( dom  F  \  { x } ) ) )
4842, 47fveq12d 5736 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  =  ( K `  ( F
" ( dom  F  \  { x } ) ) ) )
4938, 48eleq12d 2506 . . . . . . . 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 287 . . . . . . 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 2918 . . . . . 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 2918 . . . . 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 250 . . . 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 693 . . 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 27255 . . 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 4471 . 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 441 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   [.wsbc 3163    \ cdif 3319   {csn 3816   class class class wbr 4214   dom cdm 4880   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471  Scalarcsca 13534   .scvsca 13535   0gc0g 13725   LSpanclspn 16049   LIndF clindf 27253
This theorem is referenced by:  islinds2  27262  islindf2  27263  lindff  27264  lindfind  27265  f1lindf  27271  lsslindf  27279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-lindf 27255
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