Definition df-lindf 21924

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 21944, 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 21956) and only one representation for each element of the range (islindf5 21957). (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 21922	. 2 class LIndF
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1566	. . . . . 6 class 𝑓
4	3	cdm 5662	. . . . 5 class dom 𝑓
5		vw	. . . . . . 7 setvar 𝑤
6	5	cv 1566	. . . . . 6 class 𝑤
7		cbs 17268	. . . . . 6 class Base
8	6, 7	cfv 6537	. . . . 5 class (Base‘𝑤)
9	4, 8, 3	wf 6533	. . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10		vk	. . . . . . . . . . 11 setvar 𝑘
11	10	cv 1566	. . . . . . . . . 10 class 𝑘
12		vx	. . . . . . . . . . . 12 setvar 𝑥
13	12	cv 1566	. . . . . . . . . . 11 class 𝑥
14	13, 3	cfv 6537	. . . . . . . . . 10 class (𝑓‘𝑥)
15		cvsca 17313	. . . . . . . . . . 11 class ·𝑠
16	6, 15	cfv 6537	. . . . . . . . . 10 class ( ·𝑠 ‘𝑤)
17	11, 14, 16	co 7411	. . . . . . . . 9 class (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥))
18	13	csn 4594	. . . . . . . . . . . 12 class {𝑥}
19	4, 18	cdif 3910	. . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
20	3, 19	cima 5665	. . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21		clspn 21069	. . . . . . . . . . 11 class LSpan
22	6, 21	cfv 6537	. . . . . . . . . 10 class (LSpan‘𝑤)
23	20, 22	cfv 6537	. . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
24	17, 23	wcel 2149	. . . . . . . 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 1566	. . . . . . . . 9 class 𝑠
28	27, 7	cfv 6537	. . . . . . . 8 class (Base‘𝑠)
29		c0g 17491	. . . . . . . . . 10 class 0g
30	27, 29	cfv 6537	. . . . . . . . 9 class (0g‘𝑠)
31	30	csn 4594	. . . . . . . 8 class {(0g‘𝑠)}
32	28, 31	cdif 3910	. . . . . . 7 class ((Base‘𝑠) ∖ {(0g‘𝑠)})
33	25, 10, 32	wral 3085	. . . . . 6 wff ∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
34	33, 12, 4	wral 3085	. . . . 5 wff ∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35		csca 17312	. . . . . 6 class Scalar
36	6, 35	cfv 6537	. . . . 5 class (Scalar‘𝑤)
37	34, 26, 36	wsbc 3753	. . . 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 5177	. 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 1567	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  21926  islindf  21930