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Theorem lindff 27253
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 444 . . 3  |-  ( ( F LIndF  W  /\  W  e.  Y )  ->  F LIndF  W )
2 rellindf 27246 . . . . . 6  |-  Rel LIndF
32brrelexi 4910 . . . . 5  |-  ( F LIndF 
W  ->  F  e.  _V )
4 lindff.b . . . . . 6  |-  B  =  ( Base `  W
)
5 eqid 2435 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
6 eqid 2435 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
7 eqid 2435 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2435 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
9 eqid 2435 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 8, 9islindf 27250 . . . . 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 461 . . . 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 440 . . 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 202 . 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 446 1  |-  ( ( F LIndF  W  /\  W  e.  Y )  ->  F : dom  F --> B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    \ cdif 3309   {csn 3806   class class class wbr 4204   dom cdm 4870   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   LSpanclspn 16039   LIndF clindf 27242
This theorem is referenced by:  lindfind2  27256  lindff1  27258  lindfrn  27259  f1lindf  27260  indlcim  27278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-lindf 27244
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