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Theorem islindf2 27387
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 955 . . 3  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  W  e.  Y
)
2 simp3 957 . . . 4  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  F : I --> B )
3 simp2 956 . . . 4  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  I  e.  X
)
4 fex 5765 . . . 4  |-  ( ( F : I --> B  /\  I  e.  X )  ->  F  e.  _V )
52, 3, 4syl2anc 642 . . 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 27385 . . 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 642 . 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 5419 . . . . 5  |-  ( F : I --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  I ) )
1514simpld 445 . . . 4  |-  ( F : I --> B  ->  F : dom  F --> B )
16153ad2ant3 978 . . 3  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  F : dom  F --> B )
1716biantrurd 494 . 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 5409 . . . 4  |-  ( F : I --> B  ->  dom  F  =  I )
19183ad2ant3 978 . . 3  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  dom  F  =  I )
2019difeq1d 3306 . . . . . . . 8  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( dom  F  \  { x } )  =  ( I  \  { x } ) )
2120imaeq2d 5028 . . . . . . 7  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( F "
( dom  F  \  {
x } ) )  =  ( F "
( I  \  {
x } ) ) )
2221fveq2d 5545 . . . . . 6  |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B )  ->  ( K `  ( F " ( dom 
F  \  { x } ) ) )  =  ( K `  ( F " ( I 
\  { x }
) ) ) )
2322eleq2d 2363 . . . . 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 285 . . . 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 2576 . . 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 2761 . 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 272 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {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:  lindfmm  27400  islindf4  27411
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-lindf 27379
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