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Definition df-lindf 19906
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 19926, 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 19938) and only one representation for each element of the range (islindf5 19939). (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 19904 . 2 class LIndF
2 vf . . . . . . 7 setvar 𝑓
32cv 1473 . . . . . 6 class 𝑓
43cdm 5028 . . . . 5 class dom 𝑓
5 vw . . . . . . 7 setvar 𝑤
65cv 1473 . . . . . 6 class 𝑤
7 cbs 15641 . . . . . 6 class Base
86, 7cfv 5790 . . . . 5 class (Base‘𝑤)
94, 8, 3wf 5786 . . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10 vk . . . . . . . . . . 11 setvar 𝑘
1110cv 1473 . . . . . . . . . 10 class 𝑘
12 vx . . . . . . . . . . . 12 setvar 𝑥
1312cv 1473 . . . . . . . . . . 11 class 𝑥
1413, 3cfv 5790 . . . . . . . . . 10 class (𝑓𝑥)
15 cvsca 15718 . . . . . . . . . . 11 class ·𝑠
166, 15cfv 5790 . . . . . . . . . 10 class ( ·𝑠𝑤)
1711, 14, 16co 6527 . . . . . . . . 9 class (𝑘( ·𝑠𝑤)(𝑓𝑥))
1813csn 4124 . . . . . . . . . . . 12 class {𝑥}
194, 18cdif 3536 . . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
203, 19cima 5031 . . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21 clspn 18738 . . . . . . . . . . 11 class LSpan
226, 21cfv 5790 . . . . . . . . . 10 class (LSpan‘𝑤)
2320, 22cfv 5790 . . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
2417, 23wcel 1976 . . . . . . . 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 1473 . . . . . . . . 9 class 𝑠
2827, 7cfv 5790 . . . . . . . 8 class (Base‘𝑠)
29 c0g 15869 . . . . . . . . . 10 class 0g
3027, 29cfv 5790 . . . . . . . . 9 class (0g𝑠)
3130csn 4124 . . . . . . . 8 class {(0g𝑠)}
3228, 31cdif 3536 . . . . . . 7 class ((Base‘𝑠) ∖ {(0g𝑠)})
3325, 10, 32wral 2895 . . . . . 6 wff 𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
3433, 12, 4wral 2895 . . . . 5 wff 𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35 csca 15717 . . . . . 6 class Scalar
366, 35cfv 5790 . . . . 5 class (Scalar‘𝑤)
3734, 26, 36wsbc 3401 . . . 4 wff [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
389, 37wa 382 . . 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 4636 . 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 1474 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  19908  islindf  19912
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