MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  islindf Structured version   Visualization version   Unicode version

Theorem islindf 19382
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 5715 . . . . . 6  |-  ( f  =  F  ->  (
f : dom  f --> ( Base `  w )  <->  F : dom  f --> (
Base `  w )
) )
21adantr 467 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f : dom  f
--> ( Base `  w
)  <->  F : dom  f --> ( Base `  w )
) )
3 dmeq 5038 . . . . . . 7  |-  ( f  =  F  ->  dom  f  =  dom  F )
43adantr 467 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  dom  f  =  dom  F )
5 fveq2 5870 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
6 islindf.b . . . . . . . 8  |-  B  =  ( Base `  W
)
75, 6syl6eqr 2505 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  B )
87adantl 468 . . . . . 6  |-  ( ( f  =  F  /\  w  =  W )  ->  ( Base `  w
)  =  B )
94, 8feq23d 5728 . . . . 5  |-  ( ( f  =  F  /\  w  =  W )  ->  ( F : dom  f
--> ( Base `  w
)  <->  F : dom  F --> B ) )
102, 9bitrd 257 . . . 4  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f : dom  f
--> ( Base `  w
)  <->  F : dom  F --> B ) )
11 fvex 5880 . . . . . 6  |-  (Scalar `  w )  e.  _V
12 fveq2 5870 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  ( Base `  s )  =  (
Base `  (Scalar `  w
) ) )
13 fveq2 5870 . . . . . . . . . 10  |-  ( s  =  (Scalar `  w
)  ->  ( 0g `  s )  =  ( 0g `  (Scalar `  w ) ) )
1413sneqd 3982 . . . . . . . . 9  |-  ( s  =  (Scalar `  w
)  ->  { ( 0g `  s ) }  =  { ( 0g
`  (Scalar `  w )
) } )
1512, 14difeq12d 3554 . . . . . . . 8  |-  ( s  =  (Scalar `  w
)  ->  ( ( Base `  s )  \  { ( 0g `  s ) } )  =  ( ( Base `  (Scalar `  w )
)  \  { ( 0g `  (Scalar `  w
) ) } ) )
1615raleqdv 2995 . . . . . . 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 2829 . . . . . 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 3304 . . . . 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 5870 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
20 islindf.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
2119, 20syl6eqr 2505 . . . . . . . . . . 11  |-  ( w  =  W  ->  (Scalar `  w )  =  S )
2221fveq2d 5874 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  S )
)
23 islindf.n . . . . . . . . . 10  |-  N  =  ( Base `  S
)
2422, 23syl6eqr 2505 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  N )
2521fveq2d 5874 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  S ) )
26 islindf.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  S )
2725, 26syl6eqr 2505 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
2827sneqd 3982 . . . . . . . . 9  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
2924, 28difeq12d 3554 . . . . . . . 8  |-  ( w  =  W  ->  (
( Base `  (Scalar `  w
) )  \  {
( 0g `  (Scalar `  w ) ) } )  =  ( N 
\  {  .0.  }
) )
3029adantl 468 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( Base `  (Scalar `  w ) )  \  { ( 0g `  (Scalar `  w ) ) } )  =  ( N  \  {  .0.  } ) )
31 fveq2 5870 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
32 islindf.v . . . . . . . . . . . 12  |-  .x.  =  ( .s `  W )
3331, 32syl6eqr 2505 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
3433adantl 468 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( .s `  w
)  =  .x.  )
35 eqidd 2454 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  k  =  k )
36 fveq1 5869 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
3736adantr 467 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f `  x
)  =  ( F `
 x ) )
3834, 35, 37oveq123d 6316 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( k ( .s
`  w ) ( f `  x ) )  =  ( k 
.x.  ( F `  x ) ) )
39 fveq2 5870 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
40 islindf.k . . . . . . . . . . . 12  |-  K  =  ( LSpan `  W )
4139, 40syl6eqr 2505 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSpan `  w )  =  K )
4241adantl 468 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( LSpan `  w )  =  K )
43 imaeq1 5166 . . . . . . . . . . . 12  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  f  \  { x } ) ) )
443difeq1d 3552 . . . . . . . . . . . . 13  |-  ( f  =  F  ->  ( dom  f  \  { x } )  =  ( dom  F  \  {
x } ) )
4544imaeq2d 5171 . . . . . . . . . . . 12  |-  ( f  =  F  ->  ( F " ( dom  f  \  { x } ) )  =  ( F
" ( dom  F  \  { x } ) ) )
4643, 45eqtrd 2487 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f " ( dom  f  \  { x } ) )  =  ( F " ( dom  F  \  { x } ) ) )
4746adantr 467 . . . . . . . . . 10  |-  ( ( f  =  F  /\  w  =  W )  ->  ( f " ( dom  f  \  { x } ) )  =  ( F " ( dom  F  \  { x } ) ) )
4842, 47fveq12d 5876 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  W )  ->  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )  =  ( K `  ( F
" ( dom  F  \  { x } ) ) ) )
4938, 48eleq12d 2525 . . . . . . . 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 296 . . . . . . 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 3003 . . . . . 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 3003 . . . . 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 261 . . . 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 718 . . 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 19376 . . 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 4718 . 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 455 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   [.wsbc 3269    \ cdif 3403   {csn 3970   class class class wbr 4405   dom cdm 4837   "cima 4840   -->wf 5581   ` cfv 5585  (class class class)co 6295   Basecbs 15133  Scalarcsca 15205   .scvsca 15206   0gc0g 15350   LSpanclspn 18206   LIndF clindf 19374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-ov 6298  df-lindf 19376
This theorem is referenced by:  islinds2  19383  islindf2  19384  lindff  19385  lindfind  19386  f1lindf  19392  lsslindf  19400
  Copyright terms: Public domain W3C validator