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Definition df-lindf 27255
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 27275, 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 27287) and only one representation for each element of the range (islindf5 27288). (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 27253 . 2  class LIndF
2 vf . . . . . . 7  set  f
32cv 1652 . . . . . 6  class  f
43cdm 4880 . . . . 5  class  dom  f
5 vw . . . . . . 7  set  w
65cv 1652 . . . . . 6  class  w
7 cbs 13471 . . . . . 6  class  Base
86, 7cfv 5456 . . . . 5  class  ( Base `  w )
94, 8, 3wf 5452 . . . 4  wff  f : dom  f --> ( Base `  w )
10 vk . . . . . . . . . . 11  set  k
1110cv 1652 . . . . . . . . . 10  class  k
12 vx . . . . . . . . . . . 12  set  x
1312cv 1652 . . . . . . . . . . 11  class  x
1413, 3cfv 5456 . . . . . . . . . 10  class  ( f `
 x )
15 cvsca 13535 . . . . . . . . . . 11  class  .s
166, 15cfv 5456 . . . . . . . . . 10  class  ( .s
`  w )
1711, 14, 16co 6083 . . . . . . . . 9  class  ( k ( .s `  w
) ( f `  x ) )
1813csn 3816 . . . . . . . . . . . 12  class  { x }
194, 18cdif 3319 . . . . . . . . . . 11  class  ( dom  f  \  { x } )
203, 19cima 4883 . . . . . . . . . 10  class  ( f
" ( dom  f  \  { x } ) )
21 clspn 16049 . . . . . . . . . . 11  class  LSpan
226, 21cfv 5456 . . . . . . . . . 10  class  ( LSpan `  w )
2320, 22cfv 5456 . . . . . . . . 9  class  ( (
LSpan `  w ) `  ( f " ( dom  f  \  { x } ) ) )
2417, 23wcel 1726 . . . . . . . 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 1652 . . . . . . . . 9  class  s
2827, 7cfv 5456 . . . . . . . 8  class  ( Base `  s )
29 c0g 13725 . . . . . . . . . 10  class  0g
3027, 29cfv 5456 . . . . . . . . 9  class  ( 0g
`  s )
3130csn 3816 . . . . . . . 8  class  { ( 0g `  s ) }
3228, 31cdif 3319 . . . . . . 7  class  ( (
Base `  s )  \  { ( 0g `  s ) } )
3325, 10, 32wral 2707 . . . . . 6  wff  A. k  e.  ( ( Base `  s
)  \  { ( 0g `  s ) } )  -.  ( k ( .s `  w
) ( f `  x ) )  e.  ( ( LSpan `  w
) `  ( f " ( dom  f  \  { x } ) ) )
3433, 12, 4wral 2707 . . . . 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 13534 . . . . . 6  class Scalar
366, 35cfv 5456 . . . . 5  class  (Scalar `  w )
3734, 26, 36wsbc 3163 . . . 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 360 . . 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 4267 . 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 1653 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  27257  islindf  27261
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