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Definition df-lindf 19375
Description: An independent family is a family of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements of the family. This is almost, but not quite, the same as a function into an independent set.

This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power.

We can almost define family independence as a family of unequal elements with independent range, as islindf3 19395, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring.

This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 19407) and only one representation for each element of the range (islindf5 19408). (Contributed by Stefan O'Rear, 24-Feb-2015.)

Assertion
Ref Expression
df-lindf  |- 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 } ) ) ) ) }
Distinct variable group:    w, f, s, x, k

Detailed syntax breakdown of Definition df-lindf
StepHypRef Expression
1 clindf 19373 . 2  class LIndF
2 vf . . . . . . 7  setvar  f
32cv 1447 . . . . . 6  class  f
43cdm 4812 . . . . 5  class  dom  f
5 vw . . . . . . 7  setvar  w
65cv 1447 . . . . . 6  class  w
7 cbs 15132 . . . . . 6  class  Base
86, 7cfv 5561 . . . . 5  class  ( Base `  w )
94, 8, 3wf 5557 . . . 4  wff  f : dom  f --> ( Base `  w )
10 vk . . . . . . . . . . 11  setvar  k
1110cv 1447 . . . . . . . . . 10  class  k
12 vx . . . . . . . . . . . 12  setvar  x
1312cv 1447 . . . . . . . . . . 11  class  x
1413, 3cfv 5561 . . . . . . . . . 10  class  ( f `
 x )
15 cvsca 15205 . . . . . . . . . . 11  class  .s
166, 15cfv 5561 . . . . . . . . . 10  class  ( .s
`  w )
1711, 14, 16co 6276 . . . . . . . . 9  class  ( k ( .s `  w
) ( f `  x ) )
1813csn 3936 . . . . . . . . . . . 12  class  { x }
194, 18cdif 3369 . . . . . . . . . . 11  class  ( dom  f  \  { x } )
203, 19cima 4815 . . . . . . . . . 10  class  ( f
" ( dom  f  \  { x } ) )
21 clspn 18205 . . . . . . . . . . 11  class  LSpan
226, 21cfv 5561 . . . . . . . . . 10  class  ( LSpan `  w )
2320, 22cfv 5561 . . . . . . . . 9  class  ( (
LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )
2417, 23wcel 1891 . . . . . . . 8  wff  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )
2524wn 3 . . . . . . 7  wff  -.  (
k ( .s `  w ) ( f `
 x ) )  e.  ( ( LSpan `  w ) `  (
f " ( dom  f  \  { x } ) ) )
26 vs . . . . . . . . . 10  setvar  s
2726cv 1447 . . . . . . . . 9  class  s
2827, 7cfv 5561 . . . . . . . 8  class  ( Base `  s )
29 c0g 15349 . . . . . . . . . 10  class  0g
3027, 29cfv 5561 . . . . . . . . 9  class  ( 0g
`  s )
3130csn 3936 . . . . . . . 8  class  { ( 0g `  s ) }
3228, 31cdif 3369 . . . . . . 7  class  ( (
Base `  s )  \  { ( 0g `  s ) } )
3325, 10, 32wral 2737 . . . . . 6  wff  A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )
3433, 12, 4wral 2737 . . . . 5  wff  A. x  e.  dom  f A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )
35 csca 15204 . . . . . 6  class Scalar
366, 35cfv 5561 . . . . 5  class  (Scalar `  w )
3734, 26, 36wsbc 3235 . . . 4  wff  [. (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 } ) ) )
389, 37wa 375 . . 3  wff  ( 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 } ) ) ) )
3938, 2, 5copab 4432 . 2  class  { <. 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 } ) ) ) ) }
401, 39wceq 1448 1  wff 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 } ) ) ) ) }
Colors of variables: wff setvar class
This definition is referenced by:  rellindf  19377  islindf  19381
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