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Definition df-lindf 18648
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 18668, 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 18680) and only one representation for each element of the range (islindf5 18681). (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 18646 . 2  class LIndF
2 vf . . . . . . 7  setvar  f
32cv 1378 . . . . . 6  class  f
43cdm 4999 . . . . 5  class  dom  f
5 vw . . . . . . 7  setvar  w
65cv 1378 . . . . . 6  class  w
7 cbs 14493 . . . . . 6  class  Base
86, 7cfv 5588 . . . . 5  class  ( Base `  w )
94, 8, 3wf 5584 . . . 4  wff  f : dom  f --> ( Base `  w )
10 vk . . . . . . . . . . 11  setvar  k
1110cv 1378 . . . . . . . . . 10  class  k
12 vx . . . . . . . . . . . 12  setvar  x
1312cv 1378 . . . . . . . . . . 11  class  x
1413, 3cfv 5588 . . . . . . . . . 10  class  ( f `
 x )
15 cvsca 14562 . . . . . . . . . . 11  class  .s
166, 15cfv 5588 . . . . . . . . . 10  class  ( .s
`  w )
1711, 14, 16co 6285 . . . . . . . . 9  class  ( k ( .s `  w
) ( f `  x ) )
1813csn 4027 . . . . . . . . . . . 12  class  { x }
194, 18cdif 3473 . . . . . . . . . . 11  class  ( dom  f  \  { x } )
203, 19cima 5002 . . . . . . . . . 10  class  ( f
" ( dom  f  \  { x } ) )
21 clspn 17429 . . . . . . . . . . 11  class  LSpan
226, 21cfv 5588 . . . . . . . . . 10  class  ( LSpan `  w )
2320, 22cfv 5588 . . . . . . . . 9  class  ( (
LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )
2417, 23wcel 1767 . . . . . . . 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 1378 . . . . . . . . 9  class  s
2827, 7cfv 5588 . . . . . . . 8  class  ( Base `  s )
29 c0g 14698 . . . . . . . . . 10  class  0g
3027, 29cfv 5588 . . . . . . . . 9  class  ( 0g
`  s )
3130csn 4027 . . . . . . . 8  class  { ( 0g `  s ) }
3228, 31cdif 3473 . . . . . . 7  class  ( (
Base `  s )  \  { ( 0g `  s ) } )
3325, 10, 32wral 2814 . . . . . 6  wff  A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )
3433, 12, 4wral 2814 . . . . 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 14561 . . . . . 6  class Scalar
366, 35cfv 5588 . . . . 5  class  (Scalar `  w )
3734, 26, 36wsbc 3331 . . . 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 4504 . 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 1379 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  18650  islindf  18654
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