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Definition df-lindf 19008
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 19028, 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 19040) and only one representation for each element of the range (islindf5 19041). (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 19006 . 2  class LIndF
2 vf . . . . . . 7  setvar  f
32cv 1397 . . . . . 6  class  f
43cdm 4988 . . . . 5  class  dom  f
5 vw . . . . . . 7  setvar  w
65cv 1397 . . . . . 6  class  w
7 cbs 14716 . . . . . 6  class  Base
86, 7cfv 5570 . . . . 5  class  ( Base `  w )
94, 8, 3wf 5566 . . . 4  wff  f : dom  f --> ( Base `  w )
10 vk . . . . . . . . . . 11  setvar  k
1110cv 1397 . . . . . . . . . 10  class  k
12 vx . . . . . . . . . . . 12  setvar  x
1312cv 1397 . . . . . . . . . . 11  class  x
1413, 3cfv 5570 . . . . . . . . . 10  class  ( f `
 x )
15 cvsca 14788 . . . . . . . . . . 11  class  .s
166, 15cfv 5570 . . . . . . . . . 10  class  ( .s
`  w )
1711, 14, 16co 6270 . . . . . . . . 9  class  ( k ( .s `  w
) ( f `  x ) )
1813csn 4016 . . . . . . . . . . . 12  class  { x }
194, 18cdif 3458 . . . . . . . . . . 11  class  ( dom  f  \  { x } )
203, 19cima 4991 . . . . . . . . . 10  class  ( f
" ( dom  f  \  { x } ) )
21 clspn 17812 . . . . . . . . . . 11  class  LSpan
226, 21cfv 5570 . . . . . . . . . 10  class  ( LSpan `  w )
2320, 22cfv 5570 . . . . . . . . 9  class  ( (
LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )
2417, 23wcel 1823 . . . . . . . 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 1397 . . . . . . . . 9  class  s
2827, 7cfv 5570 . . . . . . . 8  class  ( Base `  s )
29 c0g 14929 . . . . . . . . . 10  class  0g
3027, 29cfv 5570 . . . . . . . . 9  class  ( 0g
`  s )
3130csn 4016 . . . . . . . 8  class  { ( 0g `  s ) }
3228, 31cdif 3458 . . . . . . 7  class  ( (
Base `  s )  \  { ( 0g `  s ) } )
3325, 10, 32wral 2804 . . . . . 6  wff  A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )
3433, 12, 4wral 2804 . . . . 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 14787 . . . . . 6  class Scalar
366, 35cfv 5570 . . . . 5  class  (Scalar `  w )
3734, 26, 36wsbc 3324 . . . 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 367 . . 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 4496 . 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 1398 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  19010  islindf  19014
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