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Theorem lindff 27285
Description: Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
lindff.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
lindff  |-  ( ( F LIndF  W  /\  W  e.  Y )  ->  F : dom  F --> B )

Proof of Theorem lindff
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( F LIndF  W  /\  W  e.  Y )  ->  F LIndF  W )
2 rellindf 27278 . . . . . 6  |-  Rel LIndF
32brrelexi 4729 . . . . 5  |-  ( F LIndF 
W  ->  F  e.  _V )
4 lindff.b . . . . . 6  |-  B  =  ( Base `  W
)
5 eqid 2283 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
6 eqid 2283 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
7 eqid 2283 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2283 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
9 eqid 2283 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 8, 9islindf 27282 . . . . 5  |-  ( ( W  e.  Y  /\  F  e.  _V )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  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 } ) ) ) ) ) )
113, 10sylan2 460 . . . 4  |-  ( ( W  e.  Y  /\  F LIndF  W )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  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 } ) ) ) ) ) )
1211ancoms 439 . . 3  |-  ( ( F LIndF  W  /\  W  e.  Y )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  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 } ) ) ) ) ) )
131, 12mpbid 201 . 2  |-  ( ( F LIndF  W  /\  W  e.  Y )  ->  ( F : dom  F --> B  /\  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 } ) ) ) ) )
1413simpld 445 1  |-  ( ( F LIndF  W  /\  W  e.  Y )  ->  F : dom  F --> B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149   {csn 3640   class class class wbr 4023   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LSpanclspn 15728   LIndF clindf 27274
This theorem is referenced by:  lindfind2  27288  lindff1  27290  lindfrn  27291  f1lindf  27292  indlcim  27310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-lindf 27276
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