Definition df-lindf 21956

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 21976, 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 21988) and only one representation for each element of the range (islindf5 21989). (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 21954	. 2 class LIndF
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1569	. . . . . 6 class 𝑓
4	3	cdm 5661	. . . . 5 class dom 𝑓
5		vw	. . . . . . 7 setvar 𝑤
6	5	cv 1569	. . . . . 6 class 𝑤
7		cbs 17264	. . . . . 6 class Base
8	6, 7	cfv 6536	. . . . 5 class (Base‘𝑤)
9	4, 8, 3	wf 6532	. . . 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 6536	. . . . . . . . . 10 class (𝑓‘𝑥)
15		cvsca 17309	. . . . . . . . . . 11 class ·𝑠
16	6, 15	cfv 6536	. . . . . . . . . 10 class ( ·𝑠 ‘𝑤)
17	11, 14, 16	co 7410	. . . . . . . . 9 class (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥))
18	13	csn 4589	. . . . . . . . . . . 12 class {𝑥}
19	4, 18	cdif 3902	. . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
20	3, 19	cima 5664	. . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21		clspn 21092	. . . . . . . . . . 11 class LSpan
22	6, 21	cfv 6536	. . . . . . . . . 10 class (LSpan‘𝑤)
23	20, 22	cfv 6536	. . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
24	17, 23	wcel 2143	. . . . . . . 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 6536	. . . . . . . 8 class (Base‘𝑠)
29		c0g 17487	. . . . . . . . . 10 class 0g
30	27, 29	cfv 6536	. . . . . . . . 9 class (0g‘𝑠)
31	30	csn 4589	. . . . . . . 8 class {(0g‘𝑠)}
32	28, 31	cdif 3902	. . . . . . 7 class ((Base‘𝑠) ∖ {(0g‘𝑠)})
33	25, 10, 32	wral 3079	. . . . . 6 wff ∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
34	33, 12, 4	wral 3079	. . . . 5 wff ∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35		csca 17308	. . . . . 6 class Scalar
36	6, 35	cfv 6536	. . . . 5 class (Scalar‘𝑤)
37	34, 26, 36	wsbc 3744	. . . 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 400	. . 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 referenced by:  rellindf  21958  islindf  21962