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Definition df-lindf 27276
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 27296, 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 27308) and only one representation for each element of the range (islindf5 27309). (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 27274 . 2  class LIndF
2 vf . . . . . . 7  set  f
32cv 1622 . . . . . 6  class  f
43cdm 4689 . . . . 5  class  dom  f
5 vw . . . . . . 7  set  w
65cv 1622 . . . . . 6  class  w
7 cbs 13148 . . . . . 6  class  Base
86, 7cfv 5255 . . . . 5  class  ( Base `  w )
94, 8, 3wf 5251 . . . 4  wff  f : dom  f --> ( Base `  w )
10 vk . . . . . . . . . . 11  set  k
1110cv 1622 . . . . . . . . . 10  class  k
12 vx . . . . . . . . . . . 12  set  x
1312cv 1622 . . . . . . . . . . 11  class  x
1413, 3cfv 5255 . . . . . . . . . 10  class  ( f `
 x )
15 cvsca 13212 . . . . . . . . . . 11  class  .s
166, 15cfv 5255 . . . . . . . . . 10  class  ( .s
`  w )
1711, 14, 16co 5858 . . . . . . . . 9  class  ( k ( .s `  w
) ( f `  x ) )
1813csn 3640 . . . . . . . . . . . 12  class  { x }
194, 18cdif 3149 . . . . . . . . . . 11  class  ( dom  f  \  { x } )
203, 19cima 4692 . . . . . . . . . 10  class  ( f
" ( dom  f  \  { x } ) )
21 clspn 15728 . . . . . . . . . . 11  class  LSpan
226, 21cfv 5255 . . . . . . . . . 10  class  ( LSpan `  w )
2320, 22cfv 5255 . . . . . . . . 9  class  ( (
LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )
2417, 23wcel 1684 . . . . . . . 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  set  s
2726cv 1622 . . . . . . . . 9  class  s
2827, 7cfv 5255 . . . . . . . 8  class  ( Base `  s )
29 c0g 13400 . . . . . . . . . 10  class  0g
3027, 29cfv 5255 . . . . . . . . 9  class  ( 0g
`  s )
3130csn 3640 . . . . . . . 8  class  { ( 0g `  s ) }
3228, 31cdif 3149 . . . . . . 7  class  ( (
Base `  s )  \  { ( 0g `  s ) } )
3325, 10, 32wral 2543 . . . . . 6  wff  A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )
3433, 12, 4wral 2543 . . . . 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 13211 . . . . . 6  class Scalar
366, 35cfv 5255 . . . . 5  class  (Scalar `  w )
3734, 26, 36wsbc 2991 . . . 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 358 . . 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 4076 . 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 1623 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 set class
This definition is referenced by:  rellindf  27278  islindf  27282
  Copyright terms: Public domain W3C validator