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Theorem List for Metamath Proof Explorer - 44501-44600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcscmatalt 44501 Alternative notation for the algebra of scalar matrices.
class ScMatALT
 
Definitiondf-dmatalt 44502* 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(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}))
 
Definitiondf-scmatalt 44503* 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(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)∀𝑖𝑛𝑗𝑛 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑟))}))
 
TheoremdmatALTval 44504* 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 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
 
TheoremdmatALTbas 44505* 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 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
 
TheoremdmatALTbasel 44506* 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 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐷) ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))
 
Theoremdmatbas 44507 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))
 
20.41.21.2  Linear combinations

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 44510, 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 44511, so that we can show that such sets are submodules of the corresponding modules, see lincolss 44538.
Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span 44538) "In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space [or module] is the intersection of all linear subspaces which each contain every vector in that set.", and "Alternately, the span of [a set] S may be defined as the set of all finite linear combinations of elements (vectors) of S". Whereas spans are defined according to the first approach in df-lsp 19744, the set of all linear combinations as defined by df-lco 44511 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 44543.

 
Syntaxclinc 44508 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC
 
Syntaxclinco 44509 Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
class LinCo
 
Definitiondf-linc 44510* 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 (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))))
 
Definitiondf-lco 44511* 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))})
 
Theoremlincop 44512* A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
(𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
 
Theoremlincval 44513* The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
 
Theoremdflinc2 44514* 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 (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣)))))
 
Theoremlcoop 44515* 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)       ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
 
Theoremlcoval 44516* 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)       ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶𝐵 ∧ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))
 
Theoremlincfsuppcl 44517 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 = (0g𝑅)       ((𝑀 ∈ LMod ∧ (𝑉𝑊𝑉𝐵) ∧ (𝐹 ∈ (𝑆m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝐵)
 
Theoremlinccl 44518 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 ∧ 𝑉𝐵𝑆 ∈ (𝑅m 𝑉))) → (𝑆( linC ‘𝑀)𝑉) ∈ 𝐵)
 
Theoremlincval0 44519 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
(𝑀𝑋 → (∅( linC ‘𝑀)∅) = (0g𝑀))
 
Theoremlincvalsng 44520 The linear combination over a singleton. (Contributed by AV, 25-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑉𝐵𝑌𝑅) → ({⟨𝑉, 𝑌⟩} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))
 
Theoremlincvalsn 44521 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)    &    · = ( ·𝑠𝑀)    &   𝐹 = {⟨𝑉, 𝑌⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵𝑌𝑅) → (𝐹( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))
 
Theoremlincvalpr 44522 The linear combination over an unordered pair. (Contributed by AV, 16-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)    &    + = (+g𝑀)    &   𝐹 = {⟨𝑉, 𝑋⟩, ⟨𝑊, 𝑌⟩}       (((𝑀 ∈ LMod ∧ 𝑉𝑊) ∧ (𝑉𝐵𝑋𝑅) ∧ (𝑊𝐵𝑌𝑅)) → (𝐹( linC ‘𝑀){𝑉, 𝑊}) = ((𝑋 · 𝑉) + (𝑌 · 𝑊)))
 
Theoremlincval1 44523 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, (0g𝑆)⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g𝑀))
 
Theoremlcosn0 44524 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, (0g𝑆)⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵) → (𝐹 ∈ (𝑅m {𝑉}) ∧ 𝐹 finSupp (0g𝑆) ∧ (𝐹( linC ‘𝑀){𝑉}) = (0g𝑀)))
 
Theoremlincvalsc0 44525* The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉0 )       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)
 
Theoremlcoc0 44526* 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 = (0g𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉0 )    &   𝑅 = (Base‘𝑆)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅m 𝑉) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑉) = 𝑍))
 
Theoremlinc0scn0 44527* 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 = (0g𝑆)    &    1 = (1r𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑍, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)
 
Theoremlincdifsn 44528 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)    &    · = ( ·𝑠𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑅)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ (𝐹 ∈ (𝑆m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑉) = ((𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) + ((𝐹𝑋) · 𝑋)))
 
Theoremlinc1 44529* 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 = (0g𝑆)    &    1 = (1r𝑆)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)
 
Theoremlincellss 44530 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) ∧ 𝑉𝑆) → ((𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝐹 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝑆))
 
Theoremlco0 44531 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 ∅) = {(0g𝑀)})
 
Theoremlcoel0 44532 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)) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
 
Theoremlincsum 44533 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.)
+ = (+g𝑀)    &   𝑋 = (𝐴( linC ‘𝑀)𝑉)    &   𝑌 = (𝐵( linC ‘𝑀)𝑉)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    = (+g𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑋 + 𝑌) = ((𝐴f 𝐵)( linC ‘𝑀)𝑉))
 
Theoremlincscm 44534* 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.)
= ( ·𝑠𝑀)    &    · = (.r‘(Scalar‘𝑀))    &   𝑋 = (𝐴( linC ‘𝑀)𝑉)    &   𝑅 = (Base‘(Scalar‘𝑀))    &   𝐹 = (𝑥𝑉 ↦ (𝑆 · (𝐴𝑥)))       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 𝑋) = (𝐹( linC ‘𝑀)𝑉))
 
Theoremlincsumcl 44535 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.)
+ = (+g𝑀)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉))
 
Theoremlincscmcl 44536 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.)
· = ( ·𝑠𝑀)    &   𝑅 = (Base‘(Scalar‘𝑀))       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))
 
Theoremlincsumscmcl 44537 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.)
· = ( ·𝑠𝑀)    &   𝑅 = (Base‘(Scalar‘𝑀))    &    + = (+g𝑀)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶𝑅𝐷 ∈ (𝑀 LinCo 𝑉) ∧ 𝐵 ∈ (𝑀 LinCo 𝑉))) → ((𝐶 · 𝐷) + 𝐵) ∈ (𝑀 LinCo 𝑉))
 
Theoremlincolss 44538 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))
 
Theoremellcoellss 44539* 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) ∧ 𝑉𝑆) → ∀𝑥 ∈ (𝑀 LinCo 𝑉)𝑥𝑆)
 
Theoremlcoss 44540 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))
 
Theoremlspsslco 44541 Lemma for lspeqlco 44543. (Contributed by AV, 17-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉))
 
Theoremlcosslsp 44542 Lemma for lspeqlco 44543. (Contributed by AV, 20-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉))
 
Theoremlspeqlco 44543 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 19744) 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))
 
20.41.21.3  Linear independence

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 44546 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.

Usually, the linear independence is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence 44546): "In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent." Furthermore, "In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. More generally, let V be a vector space over a field K, and let {vi | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a finite family {aj | j∈J} of elements of K, all nonzero, such that ∑j∈J ajvj=0. A set X of elements of V is linearly independent if the corresponding family{x}x∈X is linearly independent".
Remark 1: There are already definitions of (linearly) independent families (df-lindf 20950) and (linearly) independent sets (df-linds 20951). These definitions are based on the principle "of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements" or (see lbsind2 19853) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 20951 and df-lininds 44546 for (linear) independence for (left) modules is shown in lindslininds 44568.
Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly independent (see df-lindeps 44548) or, according to Wikipedia, "if at least one of the vectors in the set can be defined as a linear combination of the others", see islindeps2 44587. The reversed implication is not valid for arbitrary modules (but for arbitrary vector spaces), because it requires a division by a coefficient. Therefore, the definition of Wikipedia is equivalent to our definition for (left) vector spaces (see isldepslvec2 44589) and not for (left) modules in general.

 
Syntaxclininds 44544 Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS
 
Syntaxclindeps 44545 Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS
 
Definitiondf-lininds 44546* 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) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))))}
 
Theoremrellininds 44547 The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
Rel linIndS
 
Definitiondf-lindeps 44548* Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
 
Theoremlinindsv 44549 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))
 
Theoremislininds 44550* The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑆𝑉𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
 
Theoremlinindsi 44551* The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
 
Theoremlinindslinci 44552* 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 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥𝑆 (𝐹𝑥) = 0 )
 
Theoremislinindfis 44553* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑆 ∈ Fin ∧ 𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
 
Theoremislinindfiss 44554* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀𝑊𝑆 ∈ Fin ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
 
Theoremlinindscl 44555 A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)
(𝑆 linIndS 𝑀𝑆 ∈ 𝒫 (Base‘𝑀))
 
Theoremlindepsnlininds 44556 A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
((𝑆𝑉𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
 
Theoremislindeps 44557* The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )))
 
Theoremlincext1 44558* 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸m (𝑆 ∖ {𝑋})))) → 𝐹 ∈ (𝐸m 𝑆))
 
Theoremlincext2 44559* 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 )
 
Theoremlincext3 44560* 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸m (𝑆 ∖ {𝑋}))) ∧ (𝐺 finSupp 0 ∧ (𝑌( ·𝑠𝑀)𝑋) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})))) → (𝐹( linC ‘𝑀)𝑆) = 𝑍)
 
Theoremlindslinindsimp1 44561* Implication 1 for lindslininds 44568. (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 = (0g𝑅)    &   𝑍 = (0g𝑀)       ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))))
 
Theoremlindslinindimp2lem1 44562* Lemma 1 for lindslinindsimp2 44567. (Contributed by AV, 25-Apr-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → 𝑌𝐵)
 
Theoremlindslinindimp2lem2 44563* Lemma 2 for lindslinindsimp2 44567. (Contributed by AV, 25-Apr-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → 𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})))
 
Theoremlindslinindimp2lem3 44564* Lemma 3 for lindslinindsimp2 44567. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )
 
Theoremlindslinindimp2lem4 44565* Lemma 4 for lindslinindsimp2 44567. (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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥))
 
Theoremlindslinindsimp2lem5 44566* Lemma 5 for lindslinindsimp2 44567. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))
 
Theoremlindslinindsimp2 44567* Implication 2 for lindslininds 44568. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)       ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
 
Theoremlindslininds 44568 Equivalence of definitions df-linds 20951 and df-lininds 44546 for (linear) independence for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑆𝑉𝑀 ∈ LMod) → (𝑆 linIndS 𝑀𝑆 ∈ (LIndS‘𝑀)))
 
Theoremlinds0 44569 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.)
(𝑀𝑉 → ∅ linIndS 𝑀)
 
Theoremel0ldep 44570 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) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)
 
Theoremel0ldepsnzr 44571 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) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)
 
Theoremlindsrng01 44572 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 19647), 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)
 
Theoremlindszr 44573 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)
 
Theoremsnlindsntorlem 44574* Lemma for snlindsntor 44575. (Contributed by AV, 15-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))
 
Theoremsnlindsntor 44575* 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 𝑎𝑅, 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))
 
Theoremldepsprlem 44576 Lemma for ldepspr 44577. (Contributed by AV, 16-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)    &    1 = (1r𝑅)    &   𝑁 = (invg𝑅)       ((𝑀 ∈ LMod ∧ (𝑋𝐵𝑌𝐵𝐴𝑆)) → (𝑋 = (𝐴 · 𝑌) → (( 1 · 𝑋)(+g𝑀)((𝑁𝐴) · 𝑌)) = 𝑍))
 
Theoremldepspr 44577 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → ((𝐴𝑆𝑋 = (𝐴 · 𝑌)) → {𝑋, 𝑌} linDepS 𝑀))
 
Theoremlincresunit3lem3 44578 Lemma 3 for lincresunit3 44585. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑁 = (invg𝑅)    &    · = ( ·𝑠𝑀)       (((𝑀 ∈ LMod ∧ 𝑋𝐵𝑌𝐵) ∧ 𝐴𝑈) → (((𝑁𝐴) · 𝑋) = ((𝑁𝐴) · 𝑌) ↔ 𝑋 = 𝑌))
 
Theoremlincresunitlem1 44579 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → (𝐼‘(𝑁‘(𝐹𝑋))) ∈ 𝐸)
 
Theoremlincresunitlem2 44580 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) ∧ 𝑌𝑆) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑌)) ∈ 𝐸)
 
Theoremlincresunit1 44581* 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸m (𝑆 ∖ {𝑋})))
 
Theoremlincresunit2 44582* 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp 0 )
 
Theoremlincresunit3lem1 44583* Lemma 1 for lincresunit3 44585. (Contributed by AV, 17-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)((𝐺𝑧)( ·𝑠𝑀)𝑧)) = ((𝐹𝑧)( ·𝑠𝑀)𝑧))
 
Theoremlincresunit3lem2 44584* Lemma 2 for lincresunit3 44585. (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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧)))) = ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})))
 
Theoremlincresunit3 44585* 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)
 
Theoremlincreslvec3 44586* 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 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LVec ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ≠ 0𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)
 
Theoremislindeps2 44587* 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 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑅 ∈ NzRing) → (∃𝑠𝑆𝑓 ∈ (𝐸m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) → 𝑆 linDepS 𝑀))
 
Theoremislininds2 44588* Implication of being a linearly independent 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 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑅 ∈ NzRing) → (𝑆 linIndS 𝑀 → ∀𝑠𝑆𝑓 ∈ (𝐸m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)))
 
Theoremisldepslvec2 44589* Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 44587 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LVec ∧ 𝑆 ∈ 𝒫 𝐵) → (∃𝑠𝑆𝑓 ∈ (𝐸m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ 𝑆 linDepS 𝑀))
 
Theoremlindssnlvec 44590 A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.)
((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g𝑀)) → {𝑆} linIndS 𝑀)
 
20.41.21.4  Simple left modules and the ` ZZ `-module
 
Theoremlmod1lem1 44591* Lemma 1 for lmod1 44596. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼})
 
Theoremlmod1lem2 44592* Lemma 2 for lmod1 44596. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
 
Theoremlmod1lem3 44593* Lemma 3 for lmod1 44596. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
 
Theoremlmod1lem4 44594* Lemma 4 for lmod1 44596. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))
 
Theoremlmod1lem5 44595* Lemma 5 for lmod1 44596. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)
 
Theoremlmod1 44596* The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ LMod)
 
Theoremlmod1zr 44597 The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.)
𝑅 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    &   𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), {⟨⟨𝑍, 𝐼⟩, 𝐼⟩}⟩})       ((𝐼𝑉𝑍𝑊) → 𝑀 ∈ LMod)
 
Theoremlmod1zrnlvec 44598 There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.)
𝑅 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    &   𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), {⟨⟨𝑍, 𝐼⟩, 𝐼⟩}⟩})       ((𝐼𝑉𝑍𝑊) → 𝑀 ∉ LVec)
 
Theoremlmodn0 44599 Left modules exist. (Contributed by AV, 29-Apr-2019.)
LMod ≠ ∅
 
Theoremzlmodzxzequa 44600 Example of an equation within the -module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}    &    = ( ·𝑠𝑍)    &    = (-g𝑍)    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       ((2 𝐴) (3 𝐵)) = 0
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-44955
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