MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-lindf Structured version   Visualization version   GIF version

Definition df-lindf 20339
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 20359, 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 20371) and only one representation for each element of the range (islindf5 20372). (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 20337 . 2 class LIndF
2 vf . . . . . . 7 setvar 𝑓
32cv 1623 . . . . . 6 class 𝑓
43cdm 5258 . . . . 5 class dom 𝑓
5 vw . . . . . . 7 setvar 𝑤
65cv 1623 . . . . . 6 class 𝑤
7 cbs 16051 . . . . . 6 class Base
86, 7cfv 6041 . . . . 5 class (Base‘𝑤)
94, 8, 3wf 6037 . . . 4 wff 𝑓:dom 𝑓⟶(Base‘𝑤)
10 vk . . . . . . . . . . 11 setvar 𝑘
1110cv 1623 . . . . . . . . . 10 class 𝑘
12 vx . . . . . . . . . . . 12 setvar 𝑥
1312cv 1623 . . . . . . . . . . 11 class 𝑥
1413, 3cfv 6041 . . . . . . . . . 10 class (𝑓𝑥)
15 cvsca 16139 . . . . . . . . . . 11 class ·𝑠
166, 15cfv 6041 . . . . . . . . . 10 class ( ·𝑠𝑤)
1711, 14, 16co 6805 . . . . . . . . 9 class (𝑘( ·𝑠𝑤)(𝑓𝑥))
1813csn 4313 . . . . . . . . . . . 12 class {𝑥}
194, 18cdif 3704 . . . . . . . . . . 11 class (dom 𝑓 ∖ {𝑥})
203, 19cima 5261 . . . . . . . . . 10 class (𝑓 “ (dom 𝑓 ∖ {𝑥}))
21 clspn 19165 . . . . . . . . . . 11 class LSpan
226, 21cfv 6041 . . . . . . . . . 10 class (LSpan‘𝑤)
2320, 22cfv 6041 . . . . . . . . 9 class ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
2417, 23wcel 2131 . . . . . . . 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 1623 . . . . . . . . 9 class 𝑠
2827, 7cfv 6041 . . . . . . . 8 class (Base‘𝑠)
29 c0g 16294 . . . . . . . . . 10 class 0g
3027, 29cfv 6041 . . . . . . . . 9 class (0g𝑠)
3130csn 4313 . . . . . . . 8 class {(0g𝑠)}
3228, 31cdif 3704 . . . . . . 7 class ((Base‘𝑠) ∖ {(0g𝑠)})
3325, 10, 32wral 3042 . . . . . 6 wff 𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
3433, 12, 4wral 3042 . . . . 5 wff 𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))
35 csca 16138 . . . . . 6 class Scalar
366, 35cfv 6041 . . . . 5 class (Scalar‘𝑤)
3734, 26, 36wsbc 3568 . . . 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 4856 . 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 1624 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  20341  islindf  20345
  Copyright terms: Public domain W3C validator