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 , 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 ) and only one representation for each element of the range ( islindf5 ). (Contributed by Stefan O'Rear, 24-Feb-2015)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-lindf | ⊢ LIndF = { ⟨ 𝑓 , 𝑤 ⟩ ∣ ( 𝑓 : dom 𝑓 ⟶ ( Base ‘ 𝑤 ) ∧ [ ( Scalar ‘ 𝑤 ) / 𝑠 ] ∀ 𝑥 ∈ dom 𝑓 ∀ 𝑘 ∈ ( ( Base ‘ 𝑠 ) ∖ { ( 0g ‘ 𝑠 ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) ) } |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | clindf | ⊢ LIndF | |
| 1 | vf | ⊢ 𝑓 | |
| 2 | vw | ⊢ 𝑤 | |
| 3 | 1 | cv | ⊢ 𝑓 |
| 4 | 3 | cdm | ⊢ dom 𝑓 |
| 5 | cbs | ⊢ Base | |
| 6 | 2 | cv | ⊢ 𝑤 |
| 7 | 6 5 | cfv | ⊢ ( Base ‘ 𝑤 ) |
| 8 | 4 7 3 | wf | ⊢ 𝑓 : dom 𝑓 ⟶ ( Base ‘ 𝑤 ) |
| 9 | csca | ⊢ Scalar | |
| 10 | 6 9 | cfv | ⊢ ( Scalar ‘ 𝑤 ) |
| 11 | vs | ⊢ 𝑠 | |
| 12 | vx | ⊢ 𝑥 | |
| 13 | vk | ⊢ 𝑘 | |
| 14 | 11 | cv | ⊢ 𝑠 |
| 15 | 14 5 | cfv | ⊢ ( Base ‘ 𝑠 ) |
| 16 | c0g | ⊢ 0g | |
| 17 | 14 16 | cfv | ⊢ ( 0g ‘ 𝑠 ) |
| 18 | 17 | csn | ⊢ { ( 0g ‘ 𝑠 ) } |
| 19 | 15 18 | cdif | ⊢ ( ( Base ‘ 𝑠 ) ∖ { ( 0g ‘ 𝑠 ) } ) |
| 20 | 13 | cv | ⊢ 𝑘 |
| 21 | cvsca | ⊢ ·𝑠 | |
| 22 | 6 21 | cfv | ⊢ ( ·𝑠 ‘ 𝑤 ) |
| 23 | 12 | cv | ⊢ 𝑥 |
| 24 | 23 3 | cfv | ⊢ ( 𝑓 ‘ 𝑥 ) |
| 25 | 20 24 22 | co | ⊢ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) |
| 26 | clspn | ⊢ LSpan | |
| 27 | 6 26 | cfv | ⊢ ( LSpan ‘ 𝑤 ) |
| 28 | 23 | csn | ⊢ { 𝑥 } |
| 29 | 4 28 | cdif | ⊢ ( dom 𝑓 ∖ { 𝑥 } ) |
| 30 | 3 29 | cima | ⊢ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) |
| 31 | 30 27 | cfv | ⊢ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) |
| 32 | 25 31 | wcel | ⊢ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) |
| 33 | 32 | wn | ⊢ ¬ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) |
| 34 | 33 13 19 | wral | ⊢ ∀ 𝑘 ∈ ( ( Base ‘ 𝑠 ) ∖ { ( 0g ‘ 𝑠 ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) |
| 35 | 34 12 4 | wral | ⊢ ∀ 𝑥 ∈ dom 𝑓 ∀ 𝑘 ∈ ( ( Base ‘ 𝑠 ) ∖ { ( 0g ‘ 𝑠 ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) |
| 36 | 35 11 10 | wsbc | ⊢ [ ( Scalar ‘ 𝑤 ) / 𝑠 ] ∀ 𝑥 ∈ dom 𝑓 ∀ 𝑘 ∈ ( ( Base ‘ 𝑠 ) ∖ { ( 0g ‘ 𝑠 ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) |
| 37 | 8 36 | wa | ⊢ ( 𝑓 : dom 𝑓 ⟶ ( Base ‘ 𝑤 ) ∧ [ ( Scalar ‘ 𝑤 ) / 𝑠 ] ∀ 𝑥 ∈ dom 𝑓 ∀ 𝑘 ∈ ( ( Base ‘ 𝑠 ) ∖ { ( 0g ‘ 𝑠 ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) ) |
| 38 | 37 1 2 | copab | ⊢ { ⟨ 𝑓 , 𝑤 ⟩ ∣ ( 𝑓 : dom 𝑓 ⟶ ( Base ‘ 𝑤 ) ∧ [ ( Scalar ‘ 𝑤 ) / 𝑠 ] ∀ 𝑥 ∈ dom 𝑓 ∀ 𝑘 ∈ ( ( Base ‘ 𝑠 ) ∖ { ( 0g ‘ 𝑠 ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) ) } |
| 39 | 0 38 | wceq | ⊢ LIndF = { ⟨ 𝑓 , 𝑤 ⟩ ∣ ( 𝑓 : dom 𝑓 ⟶ ( Base ‘ 𝑤 ) ∧ [ ( Scalar ‘ 𝑤 ) / 𝑠 ] ∀ 𝑥 ∈ dom 𝑓 ∀ 𝑘 ∈ ( ( Base ‘ 𝑠 ) ∖ { ( 0g ‘ 𝑠 ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑓 ‘ 𝑥 ) ) ∈ ( ( LSpan ‘ 𝑤 ) ‘ ( 𝑓 “ ( dom 𝑓 ∖ { 𝑥 } ) ) ) ) } |