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Theorem List for Metamath Proof Explorer - 40701-40800   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremdmatALTval 40701* The algebra of x diagonal matrices over a ring . (Contributed by AV, 8-Dec-2019.)
Mat                      DMatALT        s

TheoremdmatALTbas 40702* The base set of the algebra of x diagonal matrices over a ring , i.e. the set of all x diagonal matrices over the ring . (Contributed by AV, 8-Dec-2019.)
Mat                      DMatALT

TheoremdmatALTbasel 40703* An element of the base set of the algebra of x diagonal matrices over a ring , i.e. an x diagonal matrix over the ring . (Contributed by AV, 8-Dec-2019.)
Mat                      DMatALT

Theoremdmatbas 40704 The set of all x diagonal matrices over (the ring) is the base set of the algebra of x diagonal matrices over (the ring) . (Contributed by AV, 8-Dec-2019.)
Mat                      DMat        DMatALT

21.33.15.2  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 40707, 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 40708, so that we can show that such sets are submodules of the corresponding modules, see lincolss 40735.
Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span) "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 "Alternatively, 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 18273, the set of all linear combinations as defined by df-lco 40708 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 40740.

Syntaxclinc 40705 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
linC

Syntaxclinco 40706 Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
LinCo

Definitiondf-linc 40707* 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 , Scalar s and a scalar multiplication . (Contributed by AV, 29-Mar-2019.)
linC Scalar g

Definitiondf-lco 40708* 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 Scalar finSupp Scalar linC

Theoremlincop 40709* A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
linC Scalar g

Theoremlincval 40710* The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
Scalar linC g

Theoremdflinc2 40711* Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
linC Scalar g

Theoremlcoop 40712* A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar              LinCo finSupp linC

Theoremlcoval 40713* The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar              LinCo finSupp linC

Theoremlincfsuppcl 40714 A linear combination of vectors (with finite support) is a vector. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar                     finSupp linC

Theoremlinccl 40715 A linear combination of vectors is a vector. (Contributed by AV, 31-Mar-2019.)
Scalar       linC

Theoremlincval0 40716 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
linC

Theoremlincvalsng 40717 The linear combination over a singleton. (Contributed by AV, 25-May-2019.)
Scalar                     linC

Theoremlincvalsn 40718 The linear combination over a singleton. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 25-May-2019.)
Scalar                            linC

Theoremlincvalpr 40719 The linear combination over an unordered pair. (Contributed by AV, 16-Apr-2019.)
Scalar                                   linC

Theoremlincval1 40720 The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.)
Scalar                     linC

Theoremlcosn0 40721 Properties of a linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar                     finSupp linC

Theoremlincvalsc0 40722* The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)
Scalar                            linC

Theoremlcoc0 40723* Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar                                   finSupp linC

Theoremlinc0scn0 40724* 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.)
Scalar                                   linC

Theoremlincdifsn 40725 A vector is a linear combination of a set containing this vector. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar                                   finSupp linC linC

Theoremlinc1 40726* 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.)
Scalar                            linC

Theoremlincellss 40727 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.)
Scalar finSupp Scalar linC

Theoremlco0 40728 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.)
LinCo

Theoremlcoel0 40729 The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
LinCo

Theoremlincsum 40730 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.)
linC        linC        Scalar                     finSupp finSupp linC

Theoremlincscm 40731* 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.)
Scalar       linC        Scalar              finSupp Scalar linC

Theoremlincsumcl 40732 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.)
LinCo LinCo LinCo

Theoremlincscmcl 40733 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.)
Scalar       LinCo LinCo

Theoremlincsumscmcl 40734 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.)
Scalar              LinCo LinCo LinCo

Theoremlincolss 40735 According to the statement in [Lang] p. 129, the set 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 . (Contributed by AV, 12-Apr-2019.)
LinCo

Theoremellcoellss 40736* 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.)
LinCo

Theoremlcoss 40737 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.)
LinCo

Theoremlspsslco 40738 Lemma for lspeqlco 40740. (Contributed by AV, 17-Apr-2019.)
LinCo

Theoremlcosslsp 40739 Lemma for lspeqlco 40740. (Contributed by AV, 20-Apr-2019.)
LinCo

Theoremlspeqlco 40740 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 18273) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.)
LinCo

21.33.15.3  Linear independency

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 40743 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 independency is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence): "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 non-zero, 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 19441) and (linearly) independent sets (df-linds 19442). 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 18382) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 19442 and df-lininds 40743 for (linear) independency for (left) modules is shown in lindslininds 40765.
Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly indepent (see df-lindeps 40745) 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 40784. 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 with our definition for (left) vector spaces (see isldepslvec2 40786) and not for (left) modules in general.

Syntaxclininds 40741 Extend class notation with the relation between a module and its linearly independent subsets.
linIndS

Syntaxclindeps 40742 Extend class notation with the relation between a module and its linearly dependent subsets.
linDepS

Definitiondf-lininds 40743* 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 Scalar finSupp Scalar linC Scalar

Theoremrellininds 40744 The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
linIndS

Definitiondf-lindeps 40745* Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
linDepS linIndS

Theoremlinindsv 40746 The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.)
linIndS

Theoremislininds 40747* The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                     linIndS finSupp linC

Theoremlinindsi 40748* The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                     linIndS finSupp linC

Theoremlinindslinci 40749* 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.)
Scalar                     linIndS finSupp linC

Theoremislinindfis 40750* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
Scalar                     linIndS linC

Theoremislinindfiss 40751* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
Scalar                     linIndS linC

Theoremlinindscl 40752 A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)
linIndS

Theoremlindepsnlininds 40753 A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
linDepS linIndS

Theoremislindeps 40754* The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                     linDepS finSupp linC

Theoremlincext1 40755* Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.)
Scalar

Theoremlincext2 40756* Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                                          finSupp finSupp

Theoremlincext3 40757* Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                                          finSupp linC linC

Theoremlindslinindsimp1 40758* Implication 1 for lindslininds 40765. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                            finSupp linC

Theoremlindslinindimp2lem1 40759* Lemma 1 for lindslinindsimp2 40764. (Contributed by AV, 25-Apr-2019.)
Scalar

Theoremlindslinindimp2lem2 40760* Lemma 2 for lindslinindsimp2 40764. (Contributed by AV, 25-Apr-2019.)
Scalar

Theoremlindslinindimp2lem3 40761* Lemma 3 for lindslinindsimp2 40764. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                                          finSupp finSupp

Theoremlindslinindimp2lem4 40762* Lemma 4 for lindslinindsimp2 40764. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                                          finSupp linC g

Theoremlindslinindsimp2lem5 40763* Lemma 5 for lindslinindsimp2 40764. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                            finSupp linC finSupp linC

Theoremlindslinindsimp2 40764* Implication 2 for lindslininds 40765. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                            finSupp linC

Theoremlindslininds 40765 Equivalence of definitions df-linds 19442 and df-lininds 40743 for (linear) independency for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
linIndS LIndS

Theoremlinds0 40766 The empty set is always a linearly independet subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
linIndS

Theoremel0ldep 40767 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.)
Scalar linDepS

Theoremel0ldepsnzr 40768 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.)
Scalar NzRing linDepS

Theoremlindsrng01 40769 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 18182), 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.)
Scalar              linIndS

Theoremlindszr 40770 Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.)
Scalar NzRing linIndS

Theoremsnlindsntorlem 40771* Lemma for snlindsntor 40772. (Contributed by AV, 15-Apr-2019.)
Scalar                                   linC

Theoremsnlindsntor 40772* 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., . In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero 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 , , such that . 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.)
Scalar                                   linIndS

Theoremldepsprlem 40773 Lemma for ldepspr 40774. (Contributed by AV, 16-Apr-2019.)
Scalar

Theoremldepspr 40774 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.)
Scalar                                   linDepS

Theoremlincresunit3lem3 40775 Lemma 3 for lincresunit3 40782. (Contributed by AV, 18-May-2019.)
Scalar              Unit

Theoremlincresunitlem1 40776 Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
Scalar              Unit

Theoremlincresunitlem2 40777 Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
Scalar              Unit

Theoremlincresunit1 40778* Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
Scalar              Unit

Theoremlincresunit2 40779* Property 2 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
Scalar              Unit                                                 finSupp finSupp

Theoremlincresunit3lem1 40780* Lemma 1 for lincresunit3 40782. (Contributed by AV, 17-May-2019.)
Scalar              Unit

Theoremlincresunit3lem2 40781* Lemma 2 for lincresunit3 40782. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar              Unit                                                 finSupp g linC

Theoremlincresunit3 40782* 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.)
Scalar              Unit                                                 finSupp linC linC

Theoremlincreslvec3 40783* 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.)
Scalar              Unit                                                 finSupp linC linC

Theoremislindeps2 40784* 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.)
Scalar                     NzRing finSupp linC linDepS

Theoremislininds2 40785* 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.)
Scalar                     NzRing linIndS finSupp linC

Theoremisldepslvec2 40786* Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 40784 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.)
Scalar                     finSupp linC linDepS

Theoremlindssnlvec 40787 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.)
linIndS

21.33.15.4  Simple left modules and the ` ZZ `-module

Theoremlmod1lem1 40788* Lemma 1 for lmod1 40793. (Contributed by AV, 28-Apr-2019.)
Scalar

Theoremlmod1lem2 40789* Lemma 2 for lmod1 40793. (Contributed by AV, 28-Apr-2019.)
Scalar

Theoremlmod1lem3 40790* Lemma 3 for lmod1 40793. (Contributed by AV, 29-Apr-2019.)
Scalar        Scalar

Theoremlmod1lem4 40791* Lemma 4 for lmod1 40793. (Contributed by AV, 29-Apr-2019.)
Scalar        Scalar

Theoremlmod1lem5 40792* Lemma 5 for lmod1 40793. (Contributed by AV, 28-Apr-2019.)
Scalar        Scalar

Theoremlmod1 40793* The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
Scalar

Theoremlmod1zr 40794 The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.)
Scalar

Theoremlmod1zrnlvec 40795 There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.)
Scalar

Theoremlmodn0 40796 Left modules exist. (Contributed by AV, 29-Apr-2019.)

Theoremzlmodzxzequa 40797 Example of an equation within the -module (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxznm 40798 Example of a linearly dependent set whose elements are not linear combinations of the others, see note in [Roman] p. 112). (Contributed by AV, 23-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxzldeplem 40799 A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxzequap 40800 Example of an equation within the -module (see example in [Roman] p. 112 for a linearly dependent set), written as a sum. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

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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 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