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Definition df-lindf 21826
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 21846, 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 21858) and only one representation for each element of the range (islindf5 21859). (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 21824 . 2 class LIndF
2 vf . . . . . . 7 setvar 𝑓
32cv 1539 . . . . . 6 class 𝑓
43cdm 5685 . . . . 5 class dom 𝑓
5 vw . . . . . . 7 setvar 𝑤
65cv 1539 . . . . . 6 class 𝑤
7 cbs 17247 . . . . . 6 class Base
86, 7cfv 6561 . . . . 5 class (Base‘𝑤)
94, 8, 3wf 6557 . . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10 vk . . . . . . . . . . 11 setvar 𝑘
1110cv 1539 . . . . . . . . . 10 class 𝑘
12 vx . . . . . . . . . . . 12 setvar 𝑥
1312cv 1539 . . . . . . . . . . 11 class 𝑥
1413, 3cfv 6561 . . . . . . . . . 10 class (𝑓𝑥)
15 cvsca 17301 . . . . . . . . . . 11 class ·𝑠
166, 15cfv 6561 . . . . . . . . . 10 class ( ·𝑠𝑤)
1711, 14, 16co 7431 . . . . . . . . 9 class (𝑘( ·𝑠𝑤)(𝑓𝑥))
1813csn 4626 . . . . . . . . . . . 12 class {𝑥}
194, 18cdif 3948 . . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
203, 19cima 5688 . . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21 clspn 20969 . . . . . . . . . . 11 class LSpan
226, 21cfv 6561 . . . . . . . . . 10 class (LSpan‘𝑤)
2320, 22cfv 6561 . . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
2417, 23wcel 2108 . . . . . . . 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 1539 . . . . . . . . 9 class 𝑠
2827, 7cfv 6561 . . . . . . . 8 class (Base‘𝑠)
29 c0g 17484 . . . . . . . . . 10 class 0g
3027, 29cfv 6561 . . . . . . . . 9 class (0g𝑠)
3130csn 4626 . . . . . . . 8 class {(0g𝑠)}
3228, 31cdif 3948 . . . . . . 7 class ((Base‘𝑠) ∖ {(0g𝑠)})
3325, 10, 32wral 3061 . . . . . 6 wff 𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
3433, 12, 4wral 3061 . . . . 5 wff 𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35 csca 17300 . . . . . 6 class Scalar
366, 35cfv 6561 . . . . 5 class (Scalar‘𝑤)
3734, 26, 36wsbc 3788 . . . 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 5205 . 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 1540 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  21828  islindf  21832
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