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Theorem lindff 27388
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 27381 . . . . . 6  |-  Rel LIndF
32brrelexi 4745 . . . . 5  |-  ( F LIndF 
W  ->  F  e.  _V )
4 lindff.b . . . . . 6  |-  B  =  ( Base `  W
)
5 eqid 2296 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
6 eqid 2296 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
7 eqid 2296 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2296 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
9 eqid 2296 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 8, 9islindf 27385 . . . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162   {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:  lindfind2  27391  lindff1  27393  lindfrn  27394  f1lindf  27395  indlcim  27413
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-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-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-opab 4094  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-fv 5279  df-ov 5877  df-lindf 27379
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