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Definition df-lindf 18361
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 18381, 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 18393) and only one representation for each element of the range (islindf5 18394). (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 18359 . 2  class LIndF
2 vf . . . . . . 7  setvar  f
32cv 1369 . . . . . 6  class  f
43cdm 4949 . . . . 5  class  dom  f
5 vw . . . . . . 7  setvar  w
65cv 1369 . . . . . 6  class  w
7 cbs 14293 . . . . . 6  class  Base
86, 7cfv 5527 . . . . 5  class  ( Base `  w )
94, 8, 3wf 5523 . . . 4  wff  f : dom  f --> ( Base `  w )
10 vk . . . . . . . . . . 11  setvar  k
1110cv 1369 . . . . . . . . . 10  class  k
12 vx . . . . . . . . . . . 12  setvar  x
1312cv 1369 . . . . . . . . . . 11  class  x
1413, 3cfv 5527 . . . . . . . . . 10  class  ( f `
 x )
15 cvsca 14362 . . . . . . . . . . 11  class  .s
166, 15cfv 5527 . . . . . . . . . 10  class  ( .s
`  w )
1711, 14, 16co 6201 . . . . . . . . 9  class  ( k ( .s `  w
) ( f `  x ) )
1813csn 3986 . . . . . . . . . . . 12  class  { x }
194, 18cdif 3434 . . . . . . . . . . 11  class  ( dom  f  \  { x } )
203, 19cima 4952 . . . . . . . . . 10  class  ( f
" ( dom  f  \  { x } ) )
21 clspn 17176 . . . . . . . . . . 11  class  LSpan
226, 21cfv 5527 . . . . . . . . . 10  class  ( LSpan `  w )
2320, 22cfv 5527 . . . . . . . . 9  class  ( (
LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )
2417, 23wcel 1758 . . . . . . . 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 1369 . . . . . . . . 9  class  s
2827, 7cfv 5527 . . . . . . . 8  class  ( Base `  s )
29 c0g 14498 . . . . . . . . . 10  class  0g
3027, 29cfv 5527 . . . . . . . . 9  class  ( 0g
`  s )
3130csn 3986 . . . . . . . 8  class  { ( 0g `  s ) }
3228, 31cdif 3434 . . . . . . 7  class  ( (
Base `  s )  \  { ( 0g `  s ) } )
3325, 10, 32wral 2799 . . . . . 6  wff  A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )
3433, 12, 4wral 2799 . . . . 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 14361 . . . . . 6  class Scalar
366, 35cfv 5527 . . . . 5  class  (Scalar `  w )
3734, 26, 36wsbc 3294 . . . 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 369 . . 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 4458 . 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 1370 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  18363  islindf  18367
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