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

Step	Hyp	Ref	Expression
1		clindf 18359	. 2 class LIndF
2		vf	. . . . . . 7 setvar f
3	2	cv 1369	. . . . . 6 class f
4	3	cdm 4949	. . . . 5 class dom f
5		vw	. . . . . . 7 setvar w
6	5	cv 1369	. . . . . 6 class w
7		cbs 14293	. . . . . 6 class Base
8	6, 7	cfv 5527	. . . . 5 class ( Base ` w )
9	4, 8, 3	wf 5523	. . . 4 wff f : dom f --> ( Base ` w )
10		vk	. . . . . . . . . . 11 setvar k
11	10	cv 1369	. . . . . . . . . 10 class k
12		vx	. . . . . . . . . . . 12 setvar x
13	12	cv 1369	. . . . . . . . . . 11 class x
14	13, 3	cfv 5527	. . . . . . . . . 10 class ( f ` x )
15		cvsca 14362	. . . . . . . . . . 11 class .s
16	6, 15	cfv 5527	. . . . . . . . . 10 class ( .s ` w )
17	11, 14, 16	co 6201	. . . . . . . . 9 class ( k ( .s ` w ) ( f ` x ) )
18	13	csn 3986	. . . . . . . . . . . 12 class { x }
19	4, 18	cdif 3434	. . . . . . . . . . 11 class ( dom f \ { x } )
20	3, 19	cima 4952	. . . . . . . . . 10 class ( f " ( dom f \ { x } ) )
21		clspn 17176	. . . . . . . . . . 11 class LSpan
22	6, 21	cfv 5527	. . . . . . . . . 10 class ( LSpan ` w )
23	20, 22	cfv 5527	. . . . . . . . 9 class ( ( LSpan ` w ) ` ( f " ( dom f \ { x } ) ) )
24	17, 23	wcel 1758	. . . . . . . 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 1369	. . . . . . . . 9 class s
28	27, 7	cfv 5527	. . . . . . . 8 class ( Base ` s )
29		c0g 14498	. . . . . . . . . 10 class 0g
30	27, 29	cfv 5527	. . . . . . . . 9 class ( 0g ` s )
31	30	csn 3986	. . . . . . . 8 class { ( 0g ` s ) }
32	28, 31	cdif 3434	. . . . . . 7 class ( ( Base ` s ) \ { ( 0g ` s ) } )
33	25, 10, 32	wral 2799	. . . . . 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 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
36	6, 35	cfv 5527	. . . . 5 class (Scalar ` w )
37	34, 26, 36	wsbc 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 } ) ) )
38	9, 37	wa 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 } ) ) ) )
39	38, 2, 5	copab 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 } ) ) ) ) }
40	1, 39	wceq 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 } ) ) ) ) }

This definition is referenced by:  rellindf  18363  islindf  18367