Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lindff Unicode version

Theorem lindff 26947
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 26940 . . . . . 6  |-  Rel LIndF
32brrelexi 4851 . . . . 5  |-  ( F LIndF 
W  ->  F  e.  _V )
4 lindff.b . . . . . 6  |-  B  =  ( Base `  W
)
5 eqid 2380 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
6 eqid 2380 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
7 eqid 2380 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2380 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
9 eqid 2380 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
104, 5, 6, 7, 8, 9islindf 26944 . . . . 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 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892    \ cdif 3253   {csn 3750   class class class wbr 4146   dom cdm 4811   "cima 4814   -->wf 5383   ` cfv 5387  (class class class)co 6013   Basecbs 13389  Scalarcsca 13452   .scvsca 13453   0gc0g 13643   LSpanclspn 15967   LIndF clindf 26936
This theorem is referenced by:  lindfind2  26950  lindff1  26952  lindfrn  26953  f1lindf  26954  indlcim  26972
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-lindf 26938
  Copyright terms: Public domain W3C validator