Definition df-lindf 19375

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 19395, 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 19407) and only one representation for each element of the range (islindf5 19408). (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

Step	Hyp	Ref	Expression
1		clindf 19373	. 2 class LIndF
2		vf	. . . . . . 7 setvar f
3	2	cv 1447	. . . . . 6 class f
4	3	cdm 4812	. . . . 5 class dom f
5		vw	. . . . . . 7 setvar w
6	5	cv 1447	. . . . . 6 class w
7		cbs 15132	. . . . . 6 class Base
8	6, 7	cfv 5561	. . . . 5 class ( Base ` w )
9	4, 8, 3	wf 5557	. . . 4 wff f : dom f --> ( Base ` w )
10		vk	. . . . . . . . . . 11 setvar k
11	10	cv 1447	. . . . . . . . . 10 class k
12		vx	. . . . . . . . . . . 12 setvar x
13	12	cv 1447	. . . . . . . . . . 11 class x
14	13, 3	cfv 5561	. . . . . . . . . 10 class ( f ` x )
15		cvsca 15205	. . . . . . . . . . 11 class .s
16	6, 15	cfv 5561	. . . . . . . . . 10 class ( .s ` w )
17	11, 14, 16	co 6276	. . . . . . . . 9 class ( k ( .s ` w ) ( f ` x ) )
18	13	csn 3936	. . . . . . . . . . . 12 class { x }
19	4, 18	cdif 3369	. . . . . . . . . . 11 class ( dom f \ { x } )
20	3, 19	cima 4815	. . . . . . . . . 10 class ( f " ( dom f \ { x } ) )
21		clspn 18205	. . . . . . . . . . 11 class LSpan
22	6, 21	cfv 5561	. . . . . . . . . 10 class ( LSpan ` w )
23	20, 22	cfv 5561	. . . . . . . . 9 class ( ( LSpan ` w ) ` ( f " ( dom f \ { x } ) ) )
24	17, 23	wcel 1891	. . . . . . . 8 wff ( k ( .s ` w ) ( f ` x ) ) e. ( ( LSpan ` w ) ` ( f " ( dom f \ { x } ) ) )
25	24	wn 3	. . . . . . 7 wff -. ( k ( .s ` w ) ( f ` x ) ) e. ( ( LSpan ` w ) ` ( f " ( dom f \ { x } ) ) )
26		vs	. . . . . . . . . 10 setvar s
27	26	cv 1447	. . . . . . . . 9 class s
28	27, 7	cfv 5561	. . . . . . . 8 class ( Base ` s )
29		c0g 15349	. . . . . . . . . 10 class 0g
30	27, 29	cfv 5561	. . . . . . . . 9 class ( 0g ` s )
31	30	csn 3936	. . . . . . . 8 class { ( 0g ` s ) }
32	28, 31	cdif 3369	. . . . . . 7 class ( ( Base ` s ) \ { ( 0g ` s ) } )
33	25, 10, 32	wral 2737	. . . . . 6 wff A. k e. ( ( Base ` s ) \ { ( 0g ` s ) } ) -. ( k ( .s ` w ) ( f ` x ) ) e. ( ( LSpan ` w ) ` ( f " ( dom f \ { x } ) ) )
34	33, 12, 4	wral 2737	. . . . 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 15204	. . . . . 6 class Scalar
36	6, 35	cfv 5561	. . . . 5 class (Scalar ` w )
37	34, 26, 36	wsbc 3235	. . . 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 } ) ) )
38	9, 37	wa 375	. . 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 } ) ) ) )
39	38, 2, 5	copab 4432	. 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 } ) ) ) ) }
40	1, 39	wceq 1448	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 } ) ) ) ) }

This definition is referenced by:  rellindf  19377  islindf  19381