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Definition df-lindf 20367
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 20387, 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 20399) and only one representation for each element of the range (islindf5 20400). (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 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
Distinct variable group:   𝑤,𝑓,𝑠,𝑥,𝑘

Detailed syntax breakdown of Definition df-lindf
StepHypRef Expression
1 clindf 20365 . 2 class LIndF
2 vf . . . . . . 7 setvar 𝑓
32cv 1631 . . . . . 6 class 𝑓
43cdm 5266 . . . . 5 class dom 𝑓
5 vw . . . . . . 7 setvar 𝑤
65cv 1631 . . . . . 6 class 𝑤
7 cbs 16079 . . . . . 6 class Base
86, 7cfv 6049 . . . . 5 class (Base‘𝑤)
94, 8, 3wf 6045 . . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10 vk . . . . . . . . . . 11 setvar 𝑘
1110cv 1631 . . . . . . . . . 10 class 𝑘
12 vx . . . . . . . . . . . 12 setvar 𝑥
1312cv 1631 . . . . . . . . . . 11 class 𝑥
1413, 3cfv 6049 . . . . . . . . . 10 class (𝑓𝑥)
15 cvsca 16167 . . . . . . . . . . 11 class ·𝑠
166, 15cfv 6049 . . . . . . . . . 10 class ( ·𝑠𝑤)
1711, 14, 16co 6814 . . . . . . . . 9 class (𝑘( ·𝑠𝑤)(𝑓𝑥))
1813csn 4321 . . . . . . . . . . . 12 class {𝑥}
194, 18cdif 3712 . . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
203, 19cima 5269 . . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21 clspn 19193 . . . . . . . . . . 11 class LSpan
226, 21cfv 6049 . . . . . . . . . 10 class (LSpan‘𝑤)
2320, 22cfv 6049 . . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
2417, 23wcel 2139 . . . . . . . 8 wff (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
2524wn 3 . . . . . . 7 wff ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
26 vs . . . . . . . . . 10 setvar 𝑠
2726cv 1631 . . . . . . . . 9 class 𝑠
2827, 7cfv 6049 . . . . . . . 8 class (Base‘𝑠)
29 c0g 16322 . . . . . . . . . 10 class 0g
3027, 29cfv 6049 . . . . . . . . 9 class (0g𝑠)
3130csn 4321 . . . . . . . 8 class {(0g𝑠)}
3228, 31cdif 3712 . . . . . . 7 class ((Base‘𝑠) ∖ {(0g𝑠)})
3325, 10, 32wral 3050 . . . . . 6 wff 𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
3433, 12, 4wral 3050 . . . . 5 wff 𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35 csca 16166 . . . . . 6 class Scalar
366, 35cfv 6049 . . . . 5 class (Scalar‘𝑤)
3734, 26, 36wsbc 3576 . . . 4 wff [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
389, 37wa 383 . . 3 wff (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))
3938, 2, 5copab 4864 . 2 class {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
401, 39wceq 1632 1 wff LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
Colors of variables: wff setvar class
This definition is referenced by:  rellindf  20369  islindf  20373
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