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Theorem List for Metamath Proof Explorer - 27101-27200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlnmlmic 27101 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 LNoeM LNoeM

Theorempwssplit0 27102* Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit1 27103* Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit2 27104* Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit3 27105* Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s                             LMHom

Theorempwssplit4 27106* Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
s                                    s        s        s        LMIso

Theoremfilnm 27107 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeM

Theorempwslnmlem0 27108 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM

Theorempwslnmlem1 27109* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM LNoeM

Theorempwslnmlem2 27110 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
s        s        s                      LNoeM       LNoeM       LNoeM

Theorempwslnm 27111 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM LNoeM

19.16.43  Direct sum of left modules

Syntaxcdsmm 27112 Class of module direct sum generator.
m

Definitiondf-dsmm 27113* The direct sum of a family of Abelian groups or left modules is the induced group structure on finite linear combinations of elements, here represented as functions with finite support. (Contributed by Stefan O'Rear, 7-Jan-2015.)
m ss

Theoremreldmdsmm 27114 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
m

Theoremdsmmval 27115* Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
s        m ss

Theoremdsmmbase 27116* Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
s        m

Theoremdsmmval2 27117 Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
m        m ss

Theoremdsmmbas2 27118* Base set of the direct sum module using the fndmin 5828 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
s              m

Theoremdsmmfi 27119 For finite products, the direct sum is just the module product. (Contributed by Stefan O'Rear, 1-Feb-2015.)
m s

Theoremdsmmelbas 27120* Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m

Theoremdsmm0cl 27121 The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m

Theoremdsmmacl 27122 The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m

Theoremprdsinvgd2 27123 Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s

Theoremdsmmsubg 27124 The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m                             SubGrp

Theoremdsmmlss 27125* The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Scalar        s              m

Theoremdsmmlmod 27126* The direct sum of a family of modules is a module. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Scalar        m

19.16.44  Free modules

Syntaxcfrlm 27127 Class of free module generator.
freeLMod

Syntaxcuvc 27128 Class of basic unit vectors for an explicit free module.
unitVec

Definitiondf-frlm 27129* The -dimensional free module over a ring is the product of -many copies of the ring with componentwise addition and multiplication. If is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod m ringLMod

Definitiondf-uvc 27130* unitVec is the unit vector in freeLMod along the axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
unitVec

Theoremfrlmval 27131 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        m ringLMod

Theoremfrlmlmod 27132 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmpws 27133 The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod               ringLMod s s

Theoremfrlmlss 27134 The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod               ringLMod s

Theoremfrlmpwsfi 27135 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        ringLMod s

Theoremfrlmsca 27136 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        Scalar

Theoremfrlm0 27137 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 27134). (Contributed by Stefan O'Rear, 4-Feb-2015.)
freeLMod

Theoremfrlmbas 27138* Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmelbas 27139 Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmrcl 27140 If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbassup 27141 Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbasmap 27142 Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbasf 27143 Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmplusgval 27144 Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
freeLMod

Theoremfrlmvscafval 27145 Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
freeLMod

Theoremfrlmvscaval 27146 Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmgsum 27147* Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.)
freeLMod                                                         g g

Theoremuvcfval 27148* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
unitVec

Theoremuvcval 27149* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvval 27150 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvvcl 27151 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvvcl2 27152 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvv1 27153 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvv0 27154 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcff 27155 Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.)
unitVec        freeLMod

Theoremuvcf1 27156 In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
unitVec        freeLMod               NzRing

Theoremuvcresum 27157 Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.)
unitVec        freeLMod                      g

Theoremfrlmsplit2 27158* Restriction is homomoprhic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod        freeLMod                             LMHom

Theoremfrlmsslss 27159* A subset of a free module obtained by restricting the support set is a submodule. is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.)
freeLMod

Theoremfrlmsslss2 27160* A subset of a free module obtained by restricting the support set is a submodule. is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.)
freeLMod

Theoremfrlmssuvc1 27161* A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.)
freeLMod        unitVec

Theoremfrlmssuvc2 27162* A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.)
freeLMod        unitVec

Theoremfrlmsslsp 27163* A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
freeLMod        unitVec

Theoremfrlmlbs 27164 The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.)
freeLMod        unitVec        LBasis

Theoremfrlmup1 27165* Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-Feb-2015.)
freeLMod                             g                      Scalar              LMHom

Theoremfrlmup2 27166* The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
freeLMod                             g                      Scalar                     unitVec

Theoremfrlmup3 27167* The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
freeLMod                             g                      Scalar

Theoremfrlmup4 27168* Universal propery of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Scalar       freeLMod        unitVec               LMHom

Theoremellspd 27169* The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Scalar                                          g

Theoremelfilspd 27170* Simplified version of ellspd 27169 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Scalar                                          g

19.16.45  Every set admits a group structure iff choice

Theoremunxpwdom3 27171* Weaker version of unxpwdom 7546 where a function is required only to be cancellative, not an injection. and are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into , each row must hit an element of ; by column injectivity, each row can be identified in at least one way by the element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
*

Theoremenfixsn 27172* Given two equipollent sets, a bijection can always be chosen which fixes a single point. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremmapfien2 27173* Equinumerousity relation for sets of finitely supported functions. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremfsuppeq 27174 Two ways of writing the support of a function with known codomain. MOVABLE SHORTEN nn0supp (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theorempwfi2f1o 27175* The pw2f1o 7204 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theorempwfi2en 27176* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theoremfrlmpwfi 27177 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
ℤ/n       freeLMod

Theoremgicabl 27178 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
𝑔

Theoremimasgim 27179 A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.)
s                             GrpIso

Theorembasfn 27180 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)

Theoremisnumbasgrplem1 27181 A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.)

Theoremharn0 27182 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
har

Theoremnuminfctb 27183 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremisnumbasgrplem2 27184 If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
har

Theoremisnumbasgrplem3 27185 Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theoremisnumbasabl 27186 A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
har

Theoremisnumbasgrp 27187 A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
har

Theoremdfacbasgrp 27188 A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.)
CHOICE

19.16.46  Independent sets and families

Syntaxclindf 27189 The class relationship of independent families in a module.
LIndF

Syntaxclinds 27190 The class generator of independent sets in a module.
LIndS

Definitiondf-lindf 27191* An independent family is a family of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements of the family. This is almost, but not quite, the same as a function into an independent set.

This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power.

We can almost define family independence as a family of unequal elements with independent range, as islindf3 27211, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring.

This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 27223) and only one representation for each element of the range (islindf5 27224). (Contributed by Stefan O'Rear, 24-Feb-2015.)

LIndF Scalar

Definitiondf-linds 27192* An independent set is a set which is independent as a family. See also islinds3 27219 and islinds4 27220. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremrellindf 27193 The independent-family predicate is a proper relation and can be used with brrelexi 4909. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF

Theoremislinds 27194 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremlinds1 27195 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS

Theoremlinds2 27196 An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremislindf 27197* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndF

Theoremislinds2 27198* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndS

Theoremislindf2 27199* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Scalar                     LIndF

Theoremlindff 27200 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF

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