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Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrlmval 26901 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        m ringLMod

Theoremfrlmlmod 26902 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmpws 26903 The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod               ringLMod s s

Theoremfrlmlss 26904 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 26905 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        ringLMod s

Theoremfrlmsca 26906 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        Scalar

Theoremfrlm0 26907 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 26904). (Contributed by Stefan O'Rear, 4-Feb-2015.)
freeLMod

Theoremfrlmbas 26908* Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmelbas 26909 Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmrcl 26910 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 26911 Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbasmap 26912 Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbasf 26913 Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmplusgval 26914 Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
freeLMod

Theoremfrlmvscafval 26915 Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
freeLMod

Theoremfrlmvscaval 26916 Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmgsum 26917* 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 26918* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
unitVec

Theoremuvcval 26919* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvval 26920 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvvcl 26921 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvvcl2 26922 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvv1 26923 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvv0 26924 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcff 26925 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 26926 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 26927 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 26928* Restriction is homomoprhic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod        freeLMod                             LMHom

Theoremfrlmsslss 26929* 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 26930* 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 26931* 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 26932* 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 26933* 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 26934 The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.)
freeLMod        unitVec        LBasis

Theoremfrlmup1 26935* 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 26936* The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
freeLMod                             g                      Scalar                     unitVec

Theoremfrlmup3 26937* 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 26938* Universal propery of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Scalar       freeLMod        unitVec               LMHom

Theoremellspd 26939* 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 26940* Simplified version of ellspd 26939 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 26941* Weaker version of unxpwdom 7504 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 26942* 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 26943* Equinumerousity relation for sets of finitely supported functions. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremfsuppeq 26944 Two ways of writing the support of a function with known codomain. MOVABLE SHORTEN nn0supp (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theorempwfi2f1o 26945* The pw2f1o 7163 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theorempwfi2en 26946* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theoremfrlmpwfi 26947 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
ℤ/n       freeLMod

Theoremgicabl 26948 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
𝑔

Theoremimasgim 26949 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 26950 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)

Theoremisnumbasgrplem1 26951 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 26952 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
har

Theoremnuminfctb 26953 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremisnumbasgrplem2 26954 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 26955 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 26956 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 26957 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 26958 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 26959 The class relationship of independent families in a module.
LIndF

Syntaxclinds 26960 The class generator of independent sets in a module.
LIndS

Definitiondf-lindf 26961* 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 26981, 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 26993) and only one representation for each element of the range (islindf5 26994). (Contributed by Stefan O'Rear, 24-Feb-2015.)

LIndF Scalar

Definitiondf-linds 26962* An independent set is a set which is independent as a family. See also islinds3 26989 and islinds4 26990. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremrellindf 26963 The independent-family predicate is a proper relation and can be used with brrelexi 4872. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF

Theoremislinds 26964 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremlinds1 26965 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS

Theoremlinds2 26966 An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremislindf 26967* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndF

Theoremislinds2 26968* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndS

Theoremislindf2 26969* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Scalar                     LIndF

Theoremlindff 26970 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF

Theoremlindfind 26971 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 26972 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 26973 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 26974 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 26975 A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar       NzRing LIndF

Theoremlindfrn 26976 The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndS

Theoremf1lindf 26977 Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndF

Theoremlindfres 26978 Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndF

Theoremlindsss 26979 Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndS

Theoremf1linds 26980 A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LIndS LIndF

Theoremislindf3 26981 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 26982 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 26983 Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LMHom LIndS LIndS

Theoremlindsmm2 26984 The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LMHom LIndS LIndS

Theoremlsslindf 26985 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 26986 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
s        LIndS LIndS

Theoremislbs4 26987 A basis is an independent spanning set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LBasis              LIndS

Theoremlbslinds 26988 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LBasis       LIndS

Theoremislinds3 26989 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 26990* 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 26991 The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LBasis       LBasis       LMIso

Theoremlmiclbs 26992 Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LBasis       LBasis       𝑚

Theoremislindf4 26993* 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 26994* A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.)
freeLMod                             g                      Scalar              LIndF

Theoremindlcim 26995* An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.)
freeLMod                                    g                      Scalar              LIndF               LMIso

Theoremlbslcic 26996 A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Scalar       LBasis       𝑚 freeLMod

Theoremlmisfree 26997* A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 16178 might be described as "every vector space is free." (Contributed by Stefan O'Rear, 26-Feb-2015.)
LBasis       Scalar       𝑚 freeLMod

19.16.48  Noetherian rings and left modules II

Syntaxclnr 26998 Extend class notation with the class of left Noetherian rings.
LNoeR

Definitiondf-lnr 26999 A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeR ringLMod LNoeM

Theoremislnr 27000 Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeR ringLMod LNoeM

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