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Definition df-lindf 21701
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 21721, 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 21733) and only one representation for each element of the range (islindf5 21734). (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 21699 . 2 class LIndF
2 vf . . . . . . 7 setvar 𝑓
32cv 1532 . . . . . 6 class 𝑓
43cdm 5669 . . . . 5 class dom 𝑓
5 vw . . . . . . 7 setvar 𝑤
65cv 1532 . . . . . 6 class 𝑤
7 cbs 17153 . . . . . 6 class Base
86, 7cfv 6537 . . . . 5 class (Base‘𝑤)
94, 8, 3wf 6533 . . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10 vk . . . . . . . . . . 11 setvar 𝑘
1110cv 1532 . . . . . . . . . 10 class 𝑘
12 vx . . . . . . . . . . . 12 setvar 𝑥
1312cv 1532 . . . . . . . . . . 11 class 𝑥
1413, 3cfv 6537 . . . . . . . . . 10 class (𝑓𝑥)
15 cvsca 17210 . . . . . . . . . . 11 class ·𝑠
166, 15cfv 6537 . . . . . . . . . 10 class ( ·𝑠𝑤)
1711, 14, 16co 7405 . . . . . . . . 9 class (𝑘( ·𝑠𝑤)(𝑓𝑥))
1813csn 4623 . . . . . . . . . . . 12 class {𝑥}
194, 18cdif 3940 . . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
203, 19cima 5672 . . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21 clspn 20818 . . . . . . . . . . 11 class LSpan
226, 21cfv 6537 . . . . . . . . . 10 class (LSpan‘𝑤)
2320, 22cfv 6537 . . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
2417, 23wcel 2098 . . . . . . . 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 1532 . . . . . . . . 9 class 𝑠
2827, 7cfv 6537 . . . . . . . 8 class (Base‘𝑠)
29 c0g 17394 . . . . . . . . . 10 class 0g
3027, 29cfv 6537 . . . . . . . . 9 class (0g𝑠)
3130csn 4623 . . . . . . . 8 class {(0g𝑠)}
3228, 31cdif 3940 . . . . . . 7 class ((Base‘𝑠) ∖ {(0g𝑠)})
3325, 10, 32wral 3055 . . . . . 6 wff 𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
3433, 12, 4wral 3055 . . . . 5 wff 𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35 csca 17209 . . . . . 6 class Scalar
366, 35cfv 6537 . . . . 5 class (Scalar‘𝑤)
3734, 26, 36wsbc 3772 . . . 4 wff [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
389, 37wa 395 . . 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 5203 . 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 1533 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  21703  islindf  21707
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