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Theorem islindf 26430
Description: Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-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
islindf  |-  ( ( W  e.  Y  /\  F  e.  X )  ->  ( 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 } ) ) ) ) ) )
Distinct variable groups:    k, F, x    k, N    k, W, x    .0. , k
Allowed substitution hints:    B( x, k)    S( x, k)    .x. ( x, k)    K( x, k)    N( x)    X( x, k)    Y( x, k)    .0. ( x)

Proof of Theorem islindf
Dummy variables  f  w  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 5412 . . . . . 6  |-  ( f  =  F  ->  (
f : dom  f --> ( Base `  w )  <->  F : dom  f --> (
Base `  w )
) )
21adantr 451 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f : dom  f
--> ( Base `  w
)  <->  F : dom  f --> ( Base `  w )
) )
3 dmeq 4916 . . . . . . 7  |-  ( f  =  F  ->  dom  f  =  dom  F )
43adantr 451 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  dom  f  =  dom  F )
5 fveq2 5563 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
6 islindf.b . . . . . . . 8  |-  B  =  ( Base `  W
)
75, 6syl6eqr 2366 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  B )
87adantl 452 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  ( Base `  w
)  =  B )
94, 8feq23d 5424 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( F : dom  f
--> ( Base `  w
)  <->  F : dom  F --> B ) )
102, 9bitrd 244 . . . 4  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f : dom  f
--> ( Base `  w
)  <->  F : dom  F --> B ) )
11 fvex 5577 . . . . . 6  |-  (Scalar `  w )  e.  _V
12 fveq2 5563 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  ( Base `  s )  =  (
Base `  (Scalar `  w
) ) )
13 fveq2 5563 . . . . . . . . . 10  |-  ( s  =  (Scalar `  w
)  ->  ( 0g `  s )  =  ( 0g `  (Scalar `  w ) ) )
1413sneqd 3687 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  { ( 0g `  s ) }  =  { ( 0g
`  (Scalar `  w )
) } )
1512, 14difeq12d 3329 . . . . . . . 8  |-  ( s  =  (Scalar `  w
)  ->  ( ( Base `  s )  \  { ( 0g `  s ) } )  =  ( ( Base `  (Scalar `  w )
)  \  { ( 0g `  (Scalar `  w
) ) } ) )
1615raleqdv 2776 . . . . . . 7  |-  ( s  =  (Scalar `  w
)  ->  ( A. k  e.  ( ( Base `  s )  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w ) ( f `  x
) )  e.  ( ( LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )  <->  A. k  e.  (
( Base `  (Scalar `  w
) )  \  {
( 0g `  (Scalar `  w ) ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) ) ) )
1716ralbidv 2597 . . . . . 6  |-  ( s  =  (Scalar `  w
)  ->  ( A. x  e.  dom  f A. k  e.  ( ( Base `  s )  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w ) ( f `  x
) )  e.  ( ( LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )  <->  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 } ) ) ) ) )
1811, 17sbcie 3059 . . . . 5  |-  ( [. (Scalar `  w )  / 
s ]. A. x  e. 
dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  <->  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 } ) ) ) )
19 fveq2 5563 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
20 islindf.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
2119, 20syl6eqr 2366 . . . . . . . . . . 11  |-  ( w  =  W  ->  (Scalar `  w )  =  S )
2221fveq2d 5567 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  S )
)
23 islindf.n . . . . . . . . . 10  |-  N  =  ( Base `  S
)
2422, 23syl6eqr 2366 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  N )
2521fveq2d 5567 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  S ) )
26 islindf.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  S )
2725, 26syl6eqr 2366 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
2827sneqd 3687 . . . . . . . . 9  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
2924, 28difeq12d 3329 . . . . . . . 8  |-  ( w  =  W  ->  (
( Base `  (Scalar `  w
) )  \  {
( 0g `  (Scalar `  w ) ) } )  =  ( N 
\  {  .0.  }
) )
3029adantl 452 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  =  ( N  \  {  .0.  } ) )
31 fveq2 5563 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
32 islindf.v . . . . . . . . . . . 12  |-  .x.  =  ( .s `  W )
3331, 32syl6eqr 2366 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
3433adantl 452 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( .s `  w
)  =  .x.  )
35 eqidd 2317 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  k  =  k )
36 fveq1 5562 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
3736adantr 451 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f `  x
)  =  ( F `
 x ) )
3834, 35, 37oveq123d 5921 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( k ( .s
`  w ) ( f `  x ) )  =  ( k 
.x.  ( F `  x ) ) )
39 fveq2 5563 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
40 islindf.k . . . . . . . . . . . 12  |-  K  =  ( LSpan `  W )
4139, 40syl6eqr 2366 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSpan `  w )  =  K )
4241adantl 452 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( LSpan `  w )  =  K )
43 imaeq1 5044 . . . . . . . . . . . 12  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  f  \  { x } ) ) )
443difeq1d 3327 . . . . . . . . . . . . 13  |-  ( f  =  F  ->  ( dom  f  \  { x } )  =  ( dom  F  \  {
x } ) )
4544imaeq2d 5049 . . . . . . . . . . . 12  |-  ( f  =  F  ->  ( F " ( dom  f  \  { x } ) )  =  ( F
" ( dom  F  \  { x } ) ) )
4643, 45eqtrd 2348 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  F  \  { x } ) ) )
4746adantr 451 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f " ( dom  f  \  { x } ) )  =  ( F " ( dom  F  \  { x } ) ) )
4842, 47fveq12d 5569 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  =  ( K `  ( F
" ( dom  F  \  { x } ) ) ) )
4938, 48eleq12d 2384 . . . . . . . 8  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( k ( .s `  w ) ( f `  x
) )  e.  ( ( LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )  <-> 
( k  .x.  ( F `  x )
)  e.  ( K `
 ( F "
( dom  F  \  {
x } ) ) ) ) )
5049notbid 285 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  W )  ->  ( -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  <->  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) )
5130, 50raleqbidv 2782 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  ( A. k  e.  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  -.  (
k ( .s `  w ) ( f `
 x ) )  e.  ( ( LSpan `  w ) `  (
f " ( dom  f  \  { x } ) ) )  <->  A. k  e.  ( N  \  {  .0.  }
)  -.  ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) )
524, 51raleqbidv 2782 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( 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 } ) ) )  <->  A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  } )  -.  ( k 
.x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) )
5318, 52syl5bb 248 . . . 4  |-  ( ( f  =  F  /\  w  =  W )  ->  ( [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  <->  A. x  e.  dom  F A. k  e.  ( N  \  {  .0.  } )  -.  (
k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } ) ) ) ) )
5410, 53anbi12d 691 . . 3  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( f : dom  f --> ( Base `  w )  /\  [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( 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 } ) ) ) ) ) )
55 df-lindf 26424 . . 3  |- LIndF  =  { <. f ,  w >.  |  ( f : dom  f
--> ( Base `  w
)  /\  [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) ) ) }
5654, 55brabga 4316 . 2  |-  ( ( F  e.  X  /\  W  e.  Y )  ->  ( 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 } ) ) ) ) ) )
5756ancoms 439 1  |-  ( ( W  e.  Y  /\  F  e.  X )  ->  ( 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 } ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   [.wsbc 3025    \ cdif 3183   {csn 3674   class class class wbr 4060   dom cdm 4726   "cima 4729   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195  Scalarcsca 13258   .scvsca 13259   0gc0g 13449   LSpanclspn 15777   LIndF clindf 26422
This theorem is referenced by:  islinds2  26431  islindf2  26432  lindff  26433  lindfind  26434  f1lindf  26440  lsslindf  26448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-lindf 26424
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