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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | coe1id 48301* | Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐼 = (1r‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (coe1‘𝐼) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 1 , 0 ))) | ||
| Theorem | coe1sclmulval 48302 | The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝑆 = ( ·𝑠 ‘𝑃) & ⊢ ∙ = (.r‘𝑃) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) → ((coe1‘(𝑌𝑆𝑍))‘𝑁) = (𝑌 · ((coe1‘𝑍)‘𝑁))) | ||
| Theorem | ply1mulgsumlem1 48303* | Lemma 1 for ply1mulgsum 48307. (Contributed by AV, 19-Oct-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐴 = (coe1‘𝐾) & ⊢ 𝐶 = (coe1‘𝐿) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ × = (.r‘𝑃) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑀) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))) | ||
| Theorem | ply1mulgsumlem2 48304* | Lemma 2 for ply1mulgsum 48307. (Contributed by AV, 19-Oct-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐴 = (coe1‘𝐾) & ⊢ 𝐶 = (coe1‘𝐿) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ × = (.r‘𝑃) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑀) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅))) | ||
| Theorem | ply1mulgsumlem3 48305* | Lemma 3 for ply1mulgsum 48307. (Contributed by AV, 20-Oct-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐴 = (coe1‘𝐾) & ⊢ 𝐶 = (coe1‘𝐿) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ × = (.r‘𝑃) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑀) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))))) finSupp (0g‘𝑅)) | ||
| Theorem | ply1mulgsumlem4 48306* | Lemma 4 for ply1mulgsum 48307. (Contributed by AV, 19-Oct-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐴 = (coe1‘𝐾) & ⊢ 𝐶 = (coe1‘𝐿) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ × = (.r‘𝑃) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑀) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃)) | ||
| Theorem | ply1mulgsum 48307* | The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐴 = (coe1‘𝐾) & ⊢ 𝐶 = (coe1‘𝐿) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ × = (.r‘𝑃) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑀) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝐾 × 𝐿) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋))))) | ||
| Theorem | evl1at0 48308 | Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.) |
| ⊢ 𝑂 = (eval1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑃) ⇒ ⊢ (𝑅 ∈ CRing → ((𝑂‘𝑍)‘ 0 ) = 0 ) | ||
| Theorem | evl1at1 48309 | Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.) |
| ⊢ 𝑂 = (eval1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (1r‘𝑃) ⇒ ⊢ (𝑅 ∈ CRing → ((𝑂‘𝐼)‘ 1 ) = 1 ) | ||
| Theorem | linply1 48310 | A term of the form 𝑥 − 𝐶 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 26204). (Contributed by AV, 3-Jul-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑋 − (𝐴‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐵) | ||
| Theorem | lineval 48311 | A term of the form 𝑥 − 𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉 − 𝐶 (part of ply1remlem 26204). (Contributed by AV, 3-Jul-2019.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑋 − (𝐴‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑉 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑂‘𝐺)‘𝑉) = (𝑉(-g‘𝑅)𝐶)) | ||
| Theorem | linevalexample 48312 | The polynomial 𝑥 − 3 over ℤ evaluated for 𝑥 = 5 results in 2. (Contributed by AV, 3-Jul-2019.) |
| ⊢ 𝑃 = (Poly1‘ℤring) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑋 = (var1‘ℤring) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑋 − (𝐴‘3)) & ⊢ 𝑂 = (eval1‘ℤring) ⇒ ⊢ ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = 2 | ||
In the following, alternative definitions for diagonal and scalar matrices are provided. These definitions define diagonal and scalar matrices as extensible structures, whereas Definitions df-dmat 22496 and df-scmat 22497 define diagonal and scalar matrices as sets. | ||
| Syntax | cdmatalt 48313 | Alternative notation for the algebra of diagonal matrices. |
| class DMatALT | ||
| Syntax | cscmatalt 48314 | Alternative notation for the algebra of scalar matrices. |
| class ScMatALT | ||
| Definition | df-dmatalt 48315* | Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.) |
| ⊢ DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))})) | ||
| Definition | df-scmatalt 48316* | Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn". (Contributed by AV, 8-Dec-2019.) |
| ⊢ ScMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑟))})) | ||
| Theorem | dmatALTval 48317* | The algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅. (Contributed by AV, 8-Dec-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMatALT 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})) | ||
| Theorem | dmatALTbas 48318* | The base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. the set of all 𝑁 x 𝑁 diagonal matrices over the ring 𝑅. (Contributed by AV, 8-Dec-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMatALT 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}) | ||
| Theorem | dmatALTbasel 48319* | An element of the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. an 𝑁 x 𝑁 diagonal matrix over the ring 𝑅. (Contributed by AV, 8-Dec-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMatALT 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐷) ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))) | ||
| Theorem | dmatbas 48320 | The set of all 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅 is the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅. (Contributed by AV, 8-Dec-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅))) | ||
According to Wikipedia ("Linear combination", 29-Mar-2019,
https://en.wikipedia.org/wiki/Linear_combination) "In mathematics, a
linear combination is an expression constructed from a set of terms by
multiplying each term by a constant and adding the results (e.g., a linear
combination of x and y would be any expression of the form ax + by, where a
and b are constants). The concept of linear combinations is central to
linear algebra and related fields of mathematics." In linear algebra, these
"terms" are "vectors" (elements from vector spaces or left modules), and the
constants are elements of the underlying field resp. ring. This corresponds
to the definition in [Lang] p. 129: "Let M be a module over a ring A and let
S be a subset of M. By a linear combination of elements of S (with
coefficients in A) one means a sum ∑x ∈S
axx where {ax} is a set of elements of A, ...". In the
definition in [Lang] p. 129, it is additionally claimed that "..., almost all
of which [elements of A] are equal to 0.". This is not necessarily required
in the following definition df-linc 48323, but it is essential if additions and
scalar multiplications of linear combinations are considered. Therefore, we
define the set of all linear combinations with finite support in df-lco 48324,
so that we can show that such sets are submodules of the corresponding
modules, see lincolss 48351.
| ||
| Syntax | clinc 48321 | Extend class notation with the operation constructing a linear combination (of vectors from a left module). |
| class linC | ||
| Syntax | clinco 48322 | Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support. |
| class LinCo | ||
| Definition | df-linc 48323* | Define the operation constructing a linear combination. Although this definition is taylored for linear combinations of vectors from left modules, it can be used for any structure having a Base, Scalar s and a scalar multiplication ·𝑠. (Contributed by AV, 29-Mar-2019.) |
| ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥))))) | ||
| Definition | df-lco 48324* | Define the operation constructing the set of all linear combinations for a set of vectors. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))}) | ||
| Theorem | lincop 48325* | A linear combination as operation. (Contributed by AV, 30-Mar-2019.) |
| ⊢ (𝑀 ∈ 𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥))))) | ||
| Theorem | lincval 48326* | The value of a linear combination. (Contributed by AV, 30-Mar-2019.) |
| ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥)))) | ||
| Theorem | dflinc2 48327* | Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.) |
| ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘f ( ·𝑠 ‘𝑚)( I ↾ 𝑣))))) | ||
| Theorem | lcoop 48328* | A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑅 = (Base‘𝑆) ⇒ ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))}) | ||
| Theorem | lcoval 48329* | The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑅 = (Base‘𝑆) ⇒ ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉))))) | ||
| Theorem | lincfsuppcl 48330 | A linear combination of vectors (with finite support) is a vector. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝑆 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑀 ∈ LMod ∧ (𝑉 ∈ 𝑊 ∧ 𝑉 ⊆ 𝐵) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝐵) | ||
| Theorem | linccl 48331 | A linear combination of vectors is a vector. (Contributed by AV, 31-Mar-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) ⇒ ⊢ ((𝑀 ∈ LMod ∧ (𝑉 ∈ Fin ∧ 𝑉 ⊆ 𝐵 ∧ 𝑆 ∈ (𝑅 ↑m 𝑉))) → (𝑆( linC ‘𝑀)𝑉) ∈ 𝐵) | ||
| Theorem | lincval0 48332 | The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.) |
| ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀)) | ||
| Theorem | lincvalsng 48333 | The linear combination over a singleton. (Contributed by AV, 25-May-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({〈𝑉, 𝑌〉} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉)) | ||
| Theorem | lincvalsn 48334 | The linear combination over a singleton. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 25-May-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ 𝐹 = {〈𝑉, 𝑌〉} ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝐹( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉)) | ||
| Theorem | lincvalpr 48335 | The linear combination over an unordered pair. (Contributed by AV, 16-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝐹 = {〈𝑉, 𝑋〉, 〈𝑊, 𝑌〉} ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ≠ 𝑊) ∧ (𝑉 ∈ 𝐵 ∧ 𝑋 ∈ 𝑅) ∧ (𝑊 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅)) → (𝐹( linC ‘𝑀){𝑉, 𝑊}) = ((𝑋 · 𝑉) + (𝑌 · 𝑊))) | ||
| Theorem | lincval1 48336 | The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ 𝐹 = {〈𝑉, (0g‘𝑆)〉} ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀)) | ||
| Theorem | lcosn0 48337 | Properties of a linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ 𝐹 = {〈𝑉, (0g‘𝑆)〉} ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑅 ↑m {𝑉}) ∧ 𝐹 finSupp (0g‘𝑆) ∧ (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀))) | ||
| Theorem | lincvalsc0 48338* | The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) | ||
| Theorem | lcoc0 48339* | Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) & ⊢ 𝑅 = (Base‘𝑆) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑉) = 𝑍)) | ||
| Theorem | linc0scn0 48340* | If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ 1 = (1r‘𝑆) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑍, 1 , 0 )) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) | ||
| Theorem | lincdifsn 48341 | A vector is a linear combination of a set containing this vector. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝑆 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑉) = ((𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) + ((𝐹‘𝑋) · 𝑋))) | ||
| Theorem | linc1 48342* | A vector is a linear combination of a set containing this vector. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ 1 = (1r‘𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋) | ||
| Theorem | lincellss 48343 | A linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑀) ∧ 𝑉 ⊆ 𝑆) → ((𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝐹 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝑆)) | ||
| Theorem | lco0 48344 | The set of empty linear combinations over a monoid is the singleton with the identity element of the monoid. (Contributed by AV, 12-Apr-2019.) |
| ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)}) | ||
| Theorem | lcoel0 48345 | The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉)) | ||
| Theorem | lincsum 48346 | The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ + = (+g‘𝑀) & ⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉) & ⊢ 𝑌 = (𝐵( linC ‘𝑀)𝑉) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ ✚ = (+g‘𝑆) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑋 + 𝑌) = ((𝐴 ∘f ✚ 𝐵)( linC ‘𝑀)𝑉)) | ||
| Theorem | lincscm 48347* | A linear combinations multiplied with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
| ⊢ ∙ = ( ·𝑠 ‘𝑀) & ⊢ · = (.r‘(Scalar‘𝑀)) & ⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉) & ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑆 · (𝐴‘𝑥))) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 ∙ 𝑋) = (𝐹( linC ‘𝑀)𝑉)) | ||
| Theorem | lincsumcl 48348 | The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.) |
| ⊢ + = (+g‘𝑀) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉)) | ||
| Theorem | lincscmcl 48349 | The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.) |
| ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)) | ||
| Theorem | lincsumscmcl 48350 | The sum of a linear combination and a multiplication of a linear combination with a scalar is a linear combination. (Contributed by AV, 11-Apr-2019.) |
| ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ 𝑅 = (Base‘(Scalar‘𝑀)) & ⊢ + = (+g‘𝑀) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉) ∧ 𝐵 ∈ (𝑀 LinCo 𝑉))) → ((𝐶 · 𝐷) + 𝐵) ∈ (𝑀 LinCo 𝑉)) | ||
| Theorem | lincolss 48351 | According to the statement in [Lang] p. 129, the set (LSubSp‘𝑀) of all linear combinations of a set of vectors V is a submodule (generated by V) of the module M. The elements of V are called generators of (LSubSp‘𝑀). (Contributed by AV, 12-Apr-2019.) |
| ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo 𝑉) ∈ (LSubSp‘𝑀)) | ||
| Theorem | ellcoellss 48352* | Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑀) ∧ 𝑉 ⊆ 𝑆) → ∀𝑥 ∈ (𝑀 LinCo 𝑉)𝑥 ∈ 𝑆) | ||
| Theorem | lcoss 48353 | A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ⊆ (𝑀 LinCo 𝑉)) | ||
| Theorem | lspsslco 48354 | Lemma for lspeqlco 48356. (Contributed by AV, 17-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉)) | ||
| Theorem | lcosslsp 48355 | Lemma for lspeqlco 48356. (Contributed by AV, 20-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉)) | ||
| Theorem | lspeqlco 48356 | Equivalence of a span of a set of vectors of a left module defined as the intersection of all linear subspaces which each contain every vector in that set (see df-lsp 20970) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = ((LSpan‘𝑀)‘𝑉)) | ||
According to the definition in [Lang] p. 129: "A subset S of a module M is said
to be linearly independent (over [the ring] A) if whenever we have a
linear combination ∑x ∈S axx which is equal to
0, then ax=0 for all x∈S." This definition does not care for
the finiteness of the set S (because the definition of a linear combination
in [Lang] p.129 does already assure that only a finite number of coefficients
can be 0 in the sum). Our definition df-lininds 48359 does also neither claim that
the subset must be finite, nor that almost all coefficients within the linear
combination are 0. If this is required, it must be explicitly stated as
precondition in the corresponding theorems. | ||
| Syntax | clininds 48357 | Extend class notation with the relation between a module and its linearly independent subsets. |
| class linIndS | ||
| Syntax | clindeps 48358 | Extend class notation with the relation between a module and its linearly dependent subsets. |
| class linDepS | ||
| Definition | df-lininds 48359* | Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ linIndS = {〈𝑠, 𝑚〉 ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))))} | ||
| Theorem | rellininds 48360 | The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.) |
| ⊢ Rel linIndS | ||
| Definition | df-lindeps 48361* | Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.) |
| ⊢ linDepS = {〈𝑠, 𝑚〉 ∣ ¬ 𝑠 linIndS 𝑚} | ||
| Theorem | linindsv 48362 | The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.) |
| ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V)) | ||
| Theorem | islininds 48363* | The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) | ||
| Theorem | linindsi 48364* | The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) | ||
| Theorem | linindslinci 48365* | The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 ) | ||
| Theorem | islinindfis 48366* | The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) | ||
| Theorem | islinindfiss 48367* | The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ Fin ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) | ||
| Theorem | linindscl 48368 | A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.) |
| ⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀)) | ||
| Theorem | lindepsnlininds 48369 | A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.) |
| ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀)) | ||
| Theorem | islindeps 48370* | The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ))) | ||
| Theorem | lincext1 48371* | Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧))) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})))) → 𝐹 ∈ (𝐸 ↑m 𝑆)) | ||
| Theorem | lincext2 48372* | Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧))) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 ) | ||
| Theorem | lincext3 48373* | Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧))) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ (𝐺 finSupp 0 ∧ (𝑌( ·𝑠 ‘𝑀)𝑋) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})))) → (𝐹( linC ‘𝑀)𝑆) = 𝑍) | ||
| Theorem | lindslinindsimp1 48374* | Implication 1 for lindslininds 48381. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.) |
| ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) ⇒ ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))) | ||
| Theorem | lindslinindimp2lem1 48375* | Lemma 1 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) |
| ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥)) & ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})) ⇒ ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝑌 ∈ 𝐵) | ||
| Theorem | lindslinindimp2lem2 48376* | Lemma 2 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) |
| ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥)) & ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})) ⇒ ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))) | ||
| Theorem | lindslinindimp2lem3 48377* | Lemma 3 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥)) & ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})) ⇒ ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 ) | ||
| Theorem | lindslinindimp2lem4 48378* | Lemma 4 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.) |
| ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥)) & ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})) ⇒ ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)) | ||
| Theorem | lindslinindsimp2lem5 48379* | Lemma 5 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) ⇒ ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 ))) | ||
| Theorem | lindslinindsimp2 48380* | Implication 2 for lindslininds 48381. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) ⇒ ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) | ||
| Theorem | lindslininds 48381 | Equivalence of definitions df-linds 21827 and df-lininds 48359 for (linear) independence for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → (𝑆 linIndS 𝑀 ↔ 𝑆 ∈ (LIndS‘𝑀))) | ||
| Theorem | linds0 48382 | The empty set is always a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ (𝑀 ∈ 𝑉 → ∅ linIndS 𝑀) | ||
| Theorem | el0ldep 48383 | A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀) | ||
| Theorem | el0ldepsnzr 48384 | A set containing the zero element of a module over a nonzero ring is always linearly dependent. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.) |
| ⊢ (((𝑀 ∈ LMod ∧ (Scalar‘𝑀) ∈ NzRing) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀) | ||
| Theorem | lindsrng01 48385 | Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 20872), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) ⇒ ⊢ ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀) | ||
| Theorem | lindszr 48386 | Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.) |
| ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → 𝑆 linIndS 𝑀) | ||
| Theorem | snlindsntorlem 48387* | Lemma for snlindsntor 48388. (Contributed by AV, 15-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝑆 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) → ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) | ||
| Theorem | snlindsntor 48388* | A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., (𝑟 · 𝑚) = 0. In an integral domain (a commutative ring without zero divisors), every nonzero element is regular, so a torsion element of a module over an integral domain is one annihilated by a nonzero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists 𝑎 ∈ 𝑅, 𝑎 ≠ 0, such that 𝑎 · 𝑥 = 0. This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝑆 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀)) | ||
| Theorem | ldepsprlem 48389 | Lemma for ldepspr 48390. (Contributed by AV, 16-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝑆 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ ((𝑀 ∈ LMod ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝐴 ∈ 𝑆)) → (𝑋 = (𝐴 · 𝑌) → (( 1 · 𝑋)(+g‘𝑀)((𝑁‘𝐴) · 𝑌)) = 𝑍)) | ||
| Theorem | ldepspr 48390 | If a vector is a scalar multiple of another vector, the (unordered pair containing the) two vectors are linearly dependent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝑆 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → ((𝐴 ∈ 𝑆 ∧ 𝑋 = (𝐴 · 𝑌)) → {𝑋, 𝑌} linDepS 𝑀)) | ||
| Theorem | lincresunit3lem3 48391 | Lemma 3 for lincresunit3 48398. (Contributed by AV, 18-May-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑀) ⇒ ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐴 ∈ 𝑈) → (((𝑁‘𝐴) · 𝑋) = ((𝑁‘𝐴) · 𝑌) ↔ 𝑋 = 𝑌)) | ||
| Theorem | lincresunitlem1 48392 | Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ⇒ ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝐼‘(𝑁‘(𝐹‘𝑋))) ∈ 𝐸) | ||
| Theorem | lincresunitlem2 48393 | Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ⇒ ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑌 ∈ 𝑆) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑌)) ∈ 𝐸) | ||
| Theorem | lincresunit1 48394* | Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ⇒ ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) | ||
| Theorem | lincresunit2 48395* | Property 2 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ⇒ ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺 finSupp 0 ) | ||
| Theorem | lincresunit3lem1 48396* | Lemma 1 for lincresunit3 48398. (Contributed by AV, 17-May-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ⇒ ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧)) = ((𝐹‘𝑧)( ·𝑠 ‘𝑀)𝑧)) | ||
| Theorem | lincresunit3lem2 48397* | Lemma 2 for lincresunit3 48398. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ⇒ ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧)))) = ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))) | ||
| Theorem | lincresunit3 48398* | Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ⇒ ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) | ||
| Theorem | lincreslvec3 48399* | Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) ⇒ ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) | ||
| Theorem | islindeps2 48400* | Conditions for being a linearly dependent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (∃𝑠 ∈ 𝑆 ∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) → 𝑆 linDepS 𝑀)) | ||
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