Definition df-lindf 22121

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 22141, 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 22153) and only one representation for each element of the range (islindf5 22154). (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 22119	. 2 class LIndF
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1568	. . . . . 6 class 𝑓
4	3	cdm 5651	. . . . 5 class dom 𝑓
5		vw	. . . . . . 7 setvar 𝑤
6	5	cv 1568	. . . . . 6 class 𝑤
7		cbs 17396	. . . . . 6 class Base
8	6, 7	cfv 6543	. . . . 5 class (Base‘𝑤)
9	4, 8, 3	wf 6539	. . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10		vk	. . . . . . . . . . 11 setvar 𝑘
11	10	cv 1568	. . . . . . . . . 10 class 𝑘
12		vx	. . . . . . . . . . . 12 setvar 𝑥
13	12	cv 1568	. . . . . . . . . . 11 class 𝑥
14	13, 3	cfv 6543	. . . . . . . . . 10 class (𝑓‘𝑥)
15		cvsca 17441	. . . . . . . . . . 11 class ·𝑠
16	6, 15	cfv 6543	. . . . . . . . . 10 class ( ·𝑠 ‘𝑤)
17	11, 14, 16	co 7424	. . . . . . . . 9 class (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥))
18	13	csn 4584	. . . . . . . . . . . 12 class {𝑥}
19	4, 18	cdif 4044	. . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
20	3, 19	cima 5654	. . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21		clspn 21255	. . . . . . . . . . 11 class LSpan
22	6, 21	cfv 6543	. . . . . . . . . 10 class (LSpan‘𝑤)
23	20, 22	cfv 6543	. . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
24	17, 23	wcel 2144	. . . . . . . 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 1568	. . . . . . . . 9 class 𝑠
28	27, 7	cfv 6543	. . . . . . . 8 class (Base‘𝑠)
29		c0g 17619	. . . . . . . . . 10 class 0g
30	27, 29	cfv 6543	. . . . . . . . 9 class (0g‘𝑠)
31	30	csn 4584	. . . . . . . 8 class {(0g‘𝑠)}
32	28, 31	cdif 4044	. . . . . . 7 class ((Base‘𝑠) ∖ {(0g‘𝑠)})
33	25, 10, 32	wral 3076	. . . . . 6 wff ∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
34	33, 12, 4	wral 3076	. . . . 5 wff ∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35		csca 17440	. . . . . 6 class Scalar
36	6, 35	cfv 6543	. . . . 5 class (Scalar‘𝑤)
37	34, 26, 36	wsbc 3738	. . . 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 5166	. 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 1569	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  22123  islindf  22127