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Theorem islindf2 26954
Description: Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-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
islindf2  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( F LIndF  W  <->  A. x  e.  I  A. k  e.  ( N  \  {  .0.  } )  -.  ( k  .x.  ( F `  x ) )  e.  ( K `
 ( F "
( I  \  {
x } ) ) ) ) )
Distinct variable groups:    k, F, x    k, N    k, W, x    .0. , k    B, k, x    k, I, x    k, X, x    k, Y, x
Allowed substitution hints:    S( x, k)    .x. ( x, k)    K( x, k)    N( x)    .0. ( x)

Proof of Theorem islindf2
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  W  e.  Y
)
2 simp3 959 . . . 4  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  F : I --> B )
3 simp2 958 . . . 4  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  I  e.  X
)
4 fex 5909 . . . 4  |-  ( ( F : I --> B  /\  I  e.  X )  ->  F  e.  _V )
52, 3, 4syl2anc 643 . . 3  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  F  e.  _V )
6 islindf.b . . . 4  |-  B  =  ( Base `  W
)
7 islindf.v . . . 4  |-  .x.  =  ( .s `  W )
8 islindf.k . . . 4  |-  K  =  ( LSpan `  W )
9 islindf.s . . . 4  |-  S  =  (Scalar `  W )
10 islindf.n . . . 4  |-  N  =  ( Base `  S
)
11 islindf.z . . . 4  |-  .0.  =  ( 0g `  S )
126, 7, 8, 9, 10, 11islindf 26952 . . 3  |-  ( ( W  e.  Y  /\  F  e.  _V )  ->  ( 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 } ) ) ) ) ) )
131, 5, 12syl2anc 643 . 2  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( 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 } ) ) ) ) ) )
14 ffdm 5546 . . . . 5  |-  ( F : I --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  I ) )
1514simpld 446 . . . 4  |-  ( F : I --> B  ->  F : dom  F --> B )
16153ad2ant3 980 . . 3  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  F : dom  F --> B )
1716biantrurd 495 . 2  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  } )  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( 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 } ) ) ) ) ) )
18 fdm 5536 . . . 4  |-  ( F : I --> B  ->  dom  F  =  I )
19183ad2ant3 980 . . 3  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  dom  F  =  I )
2019difeq1d 3408 . . . . . . . 8  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( dom  F  \  { x } )  =  ( I  \  { x } ) )
2120imaeq2d 5144 . . . . . . 7  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( F "
( dom  F  \  {
x } ) )  =  ( F "
( I  \  {
x } ) ) )
2221fveq2d 5673 . . . . . 6  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( K `  ( F " ( dom 
F  \  { x } ) ) )  =  ( K `  ( F " ( I 
\  { x }
) ) ) )
2322eleq2d 2455 . . . . 5  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) )  <->  ( k  .x.  ( F `  x
) )  e.  ( K `  ( F
" ( I  \  { x } ) ) ) ) )
2423notbid 286 . . . 4  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) )  <->  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( I  \  { x } ) ) ) ) )
2524ralbidv 2670 . . 3  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( A. k  e.  ( N  \  {  .0.  } )  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) )  <->  A. k  e.  ( N  \  {  .0.  } )  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( I  \  { x } ) ) ) ) )
2619, 25raleqbidv 2860 . 2  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  } )  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) )  <->  A. x  e.  I  A. k  e.  ( N  \  {  .0.  } )  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( I  \  { x } ) ) ) ) )
2713, 17, 263bitr2d 273 1  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( F LIndF  W  <->  A. x  e.  I  A. k  e.  ( N  \  {  .0.  } )  -.  ( k  .x.  ( F `  x ) )  e.  ( K `
 ( F "
( I  \  {
x } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   _Vcvv 2900    \ cdif 3261    C_ wss 3264   {csn 3758   class class class wbr 4154   dom cdm 4819   "cima 4822   -->wf 5391   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   0gc0g 13651   LSpanclspn 15975   LIndF clindf 26944
This theorem is referenced by:  lindfmm  26967  islindf4  26978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-lindf 26946
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