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Theorem List for Metamath Proof Explorer - 27201-27300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdsmmlss 27201* 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 27202* 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 27203 Class of free module generator.
freeLMod

Syntaxcuvc 27204 Class of basic unit vectors for an explicit free module.
unitVec

Definitiondf-frlm 27205* 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 27206* unitVec is the unit vector in freeLMod along the axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
unitVec

Theoremfrlmval 27207 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        m ringLMod

Theoremfrlmlmod 27208 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmpws 27209 The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod               ringLMod s s

Theoremfrlmlss 27210 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 27211 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        ringLMod s

Theoremfrlmsca 27212 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        Scalar

Theoremfrlm0 27213 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 27210). (Contributed by Stefan O'Rear, 4-Feb-2015.)
freeLMod

Theoremfrlmbas 27214* Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmelbas 27215 Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmrcl 27216 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 27217 Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbasmap 27218 Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbasf 27219 Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmplusgval 27220 Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
freeLMod

Theoremfrlmvscafval 27221 Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
freeLMod

Theoremfrlmvscaval 27222 Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmgsum 27223* 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 27224* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
unitVec

Theoremuvcval 27225* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvval 27226 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvvcl 27227 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvvcl2 27228 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvv1 27229 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvv0 27230 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcff 27231 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 27232 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 27233 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 27234* Restriction is homomoprhic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod        freeLMod                             LMHom

Theoremfrlmsslss 27235* 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 27236* 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 27237* 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 27238* 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 27239* 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 27240 The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.)
freeLMod        unitVec        LBasis

Theoremfrlmup1 27241* 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 27242* The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
freeLMod                             g                      Scalar                     unitVec

Theoremfrlmup3 27243* 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 27244* Universal propery of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Scalar       freeLMod        unitVec               LMHom

Theoremellspd 27245* 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 27246* Simplified version of ellspd 27245 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 27247* Weaker version of unxpwdom 7560 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 27248* 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 27249* Equinumerousity relation for sets of finitely supported functions. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremfsuppeq 27250 Two ways of writing the support of a function with known codomain. MOVABLE SHORTEN nn0supp (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theorempwfi2f1o 27251* The pw2f1o 7216 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theorempwfi2en 27252* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theoremfrlmpwfi 27253 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
ℤ/n       freeLMod

Theoremgicabl 27254 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
𝑔

Theoremimasgim 27255 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 27256 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)

Theoremisnumbasgrplem1 27257 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 27258 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
har

Theoremnuminfctb 27259 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremisnumbasgrplem2 27260 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 27261 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 27262 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 27263 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 27264 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 27265 The class relationship of independent families in a module.
LIndF

Syntaxclinds 27266 The class generator of independent sets in a module.
LIndS

Definitiondf-lindf 27267* 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 27287, 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 27299) and only one representation for each element of the range (islindf5 27300). (Contributed by Stefan O'Rear, 24-Feb-2015.)

LIndF Scalar

Definitiondf-linds 27268* An independent set is a set which is independent as a family. See also islinds3 27295 and islinds4 27296. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremrellindf 27269 The independent-family predicate is a proper relation and can be used with brrelexi 4921. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF

Theoremislinds 27270 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremlinds1 27271 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS

Theoremlinds2 27272 An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremislindf 27273* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndF

Theoremislinds2 27274* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndS

Theoremislindf2 27275* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Scalar                     LIndF

Theoremlindff 27276 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF

Theoremlindfind 27277 A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndF

Theoremlindsind 27278 A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndS

Theoremlindfind2 27279 In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar       NzRing LIndF

Theoremlindsind2 27280 In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar       NzRing LIndS

Theoremlindff1 27281 A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar       NzRing LIndF

Theoremlindfrn 27282 The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndS

Theoremf1lindf 27283 Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndF

Theoremlindfres 27284 Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndF

Theoremlindsss 27285 Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndS

Theoremf1linds 27286 A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LIndS LIndF

Theoremislindf3 27287 In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Scalar       NzRing LIndF LIndS

Theoremlindfmm 27288 Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
LMHom LIndF LIndF

Theoremlindsmm 27289 Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LMHom LIndS LIndS

Theoremlindsmm2 27290 The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LMHom LIndS LIndS

Theoremlsslindf 27291 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
s        LIndF LIndF

Theoremlsslinds 27292 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
s        LIndS LIndS

Theoremislbs4 27293 A basis is an independent spanning set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LBasis              LIndS

Theoremlbslinds 27294 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LBasis       LIndS

Theoremislinds3 27295 A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.)
s        LBasis       LIndS

Theoremislinds4 27296* A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.)
LBasis       LIndS

19.16.47  Characterization of free modules

Theoremlmimlbs 27297 The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LBasis       LBasis       LMIso

Theoremlmiclbs 27298 Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LBasis       LBasis       𝑚

Theoremislindf4 27299* A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar                            freeLMod        LIndF g

Theoremislindf5 27300* A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.)
freeLMod                             g                      Scalar              LIndF

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