Definition df-lindf 21757

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 21777, 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 21789) and only one representation for each element of the range (islindf5 21790). (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 21755	. 2 class LIndF
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1532	. . . . . 6 class 𝑓
4	3	cdm 5678	. . . . 5 class dom 𝑓
5		vw	. . . . . . 7 setvar 𝑤
6	5	cv 1532	. . . . . 6 class 𝑤
7		cbs 17183	. . . . . 6 class Base
8	6, 7	cfv 6549	. . . . 5 class (Base‘𝑤)
9	4, 8, 3	wf 6545	. . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10		vk	. . . . . . . . . . 11 setvar 𝑘
11	10	cv 1532	. . . . . . . . . 10 class 𝑘
12		vx	. . . . . . . . . . . 12 setvar 𝑥
13	12	cv 1532	. . . . . . . . . . 11 class 𝑥
14	13, 3	cfv 6549	. . . . . . . . . 10 class (𝑓‘𝑥)
15		cvsca 17240	. . . . . . . . . . 11 class ·𝑠
16	6, 15	cfv 6549	. . . . . . . . . 10 class ( ·𝑠 ‘𝑤)
17	11, 14, 16	co 7419	. . . . . . . . 9 class (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥))
18	13	csn 4630	. . . . . . . . . . . 12 class {𝑥}
19	4, 18	cdif 3941	. . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
20	3, 19	cima 5681	. . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21		clspn 20867	. . . . . . . . . . 11 class LSpan
22	6, 21	cfv 6549	. . . . . . . . . 10 class (LSpan‘𝑤)
23	20, 22	cfv 6549	. . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
24	17, 23	wcel 2098	. . . . . . . 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 1532	. . . . . . . . 9 class 𝑠
28	27, 7	cfv 6549	. . . . . . . 8 class (Base‘𝑠)
29		c0g 17424	. . . . . . . . . 10 class 0g
30	27, 29	cfv 6549	. . . . . . . . 9 class (0g‘𝑠)
31	30	csn 4630	. . . . . . . 8 class {(0g‘𝑠)}
32	28, 31	cdif 3941	. . . . . . 7 class ((Base‘𝑠) ∖ {(0g‘𝑠)})
33	25, 10, 32	wral 3050	. . . . . 6 wff ∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
34	33, 12, 4	wral 3050	. . . . 5 wff ∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35		csca 17239	. . . . . 6 class Scalar
36	6, 35	cfv 6549	. . . . 5 class (Scalar‘𝑤)
37	34, 26, 36	wsbc 3773	. . . 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 394	. . 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 5211	. 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 1533	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  21759  islindf  21763

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