Definition df-lindf 22167

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 22187, 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 22199) and only one representation for each element of the range (islindf5 22200). (Contributed by Stefan O'Rear, 24-Feb-2015.)

Assertion

Ref	Expression
df-lindf	⊢ LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((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 22165	. 2 class LIndF
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1569	. . . . . 6 class 𝑓
4	3	cdm 5658	. . . . 5 class dom 𝑓
5		vw	. . . . . . 7 setvar 𝑤
6	5	cv 1569	. . . . . 6 class 𝑤
7		cbs 17431	. . . . . 6 class Base
8	6, 7	cfv 6566	. . . . 5 class (Base‘𝑤)
9	4, 8, 3	wf 6562	. . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10		vk	. . . . . . . . . . 11 setvar 𝑘
11	10	cv 1569	. . . . . . . . . 10 class 𝑘
12		vx	. . . . . . . . . . . 12 setvar 𝑥
13	12	cv 1569	. . . . . . . . . . 11 class 𝑥
14	13, 3	cfv 6566	. . . . . . . . . 10 class (𝑓‘𝑥)
15		cvsca 17476	. . . . . . . . . . 11 class ·𝑠
16	6, 15	cfv 6566	. . . . . . . . . 10 class ( ·𝑠 ‘𝑤)
17	11, 14, 16	co 7451	. . . . . . . . 9 class (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥))
18	13	csn 4588	. . . . . . . . . . . 12 class {𝑥}
19	4, 18	cdif 4048	. . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
20	3, 19	cima 5661	. . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21		clspn 21297	. . . . . . . . . . 11 class LSpan
22	6, 21	cfv 6566	. . . . . . . . . 10 class (LSpan‘𝑤)
23	20, 22	cfv 6566	. . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
24	17, 23	wcel 2147	. . . . . . . 8 wff (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
25	24	wn 3	. . . . . . 7 wff ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
26		vs	. . . . . . . . . 10 setvar 𝑠
27	26	cv 1569	. . . . . . . . 9 class 𝑠
28	27, 7	cfv 6566	. . . . . . . 8 class (Base‘𝑠)
29		c0g 17660	. . . . . . . . . 10 class 0g
30	27, 29	cfv 6566	. . . . . . . . 9 class (0g‘𝑠)
31	30	csn 4588	. . . . . . . 8 class {(0g‘𝑠)}
32	28, 31	cdif 4048	. . . . . . 7 class ((Base‘𝑠) ∖ {(0g‘𝑠)})
33	25, 10, 32	wral 3080	. . . . . 6 wff ∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
34	33, 12, 4	wral 3080	. . . . 5 wff ∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35		csca 17475	. . . . . 6 class Scalar
36	6, 35	cfv 6566	. . . . 5 class (Scalar‘𝑤)
37	34, 26, 36	wsbc 3742	. . . 4 wff [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
38	9, 37	wa 401	. . 3 wff (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))
39	38, 2, 5	copab 5173	. 2 class {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
40	1, 39	wceq 1570	1 wff LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}

This definition is used by:  rellindf  22169  islindf  22173