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Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuvcvval 26401 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvvcl 26402 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvvcl2 26403 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvv1 26404 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcvv0 26405 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
unitVec

Theoremuvcff 26406 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 26407 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 26408 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 26409* Restriction is homomoprhic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod        freeLMod                             LMHom

Theoremfrlmsslss 26410* 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 26411* 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 26412* 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 26413* 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 26414* 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 26415 The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.)
freeLMod        unitVec        LBasis

Theoremfrlmup1 26416* 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 26417* The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
freeLMod                             g                      Scalar                     unitVec

Theoremfrlmup3 26418* 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 26419* Universal propery of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Scalar       freeLMod        unitVec               LMHom

Theoremellspd 26420* 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 26421* Simplified version of ellspd 26420 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Scalar                                          g

16.15.45  Every set admits a group structure iff choice

Theoremunxpwdom3 26422* Weaker version of unxpwdom 7187 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 26423* 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 26424* Equinumerousity relation for sets of finitely supported functions. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremfsuppeq 26425 Two ways of writing the support of a function with known codomain. MOVABLE SHORTEN nn0supp (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theorempwfi2f1o 26426* The pw2f1o 6852 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theorempwfi2en 26427* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Theoremfrlmpwfi 26428 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
ℤ/n       freeLMod

Theoremgicabl 26429 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
𝑔

Theoremimasgim 26430 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 26431 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)

Theoremisnumbasgrplem1 26432 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 26433 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
har

Theoremnuminfctb 26434 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)

Theoremisnumbasgrplem2 26435 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 26436 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 26437 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 26438 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 26439 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

16.15.46  Independent sets and families

Syntaxclindf 26440 The class relationship of independent families in a module.
LIndF

Syntaxclinds 26441 The class generator of independent sets in a module.
LIndS

Definitiondf-lindf 26442* 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 26462, 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 26474) and only one representation for each element of the range (islindf5 26475). (Contributed by Stefan O'Rear, 24-Feb-2015.)

LIndF Scalar

Definitiondf-linds 26443* An independent set is a set which is independent as a family. See also islinds3 26470 and islinds4 26471. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremrellindf 26444 The independent-family predicate is a proper relation and can be used with brrelexi 4636. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF

Theoremislinds 26445 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremlinds1 26446 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS

Theoremlinds2 26447 An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndF

Theoremislindf 26448* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndF

Theoremislinds2 26449* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar                     LIndS

Theoremislindf2 26450* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Scalar                     LIndF

Theoremlindff 26451 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF

Theoremlindfind 26452 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 26453 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 26454 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 26455 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 26456 A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Scalar       NzRing LIndF

Theoremlindfrn 26457 The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndS

Theoremf1lindf 26458 Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndF

Theoremlindfres 26459 Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndF LIndF

Theoremlindsss 26460 Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS LIndS

Theoremf1linds 26461 A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LIndS LIndF

Theoremislindf3 26462 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 26463 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 26464 Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LMHom LIndS LIndS

Theoremlindsmm2 26465 The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LMHom LIndS LIndS

Theoremlsslindf 26466 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 26467 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
s        LIndS LIndS

Theoremislbs4 26468 A basis is an independent spanning set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LBasis              LIndS

Theoremlbslinds 26469 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LBasis       LIndS

Theoremislinds3 26470 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 26471* 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

16.15.47  Characterization of free modules

Theoremlmimlbs 26472 The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LBasis       LBasis       LMIso

Theoremlmiclbs 26473 Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
LBasis       LBasis       𝑚

Theoremislindf4 26474* 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 26475* A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.)
freeLMod                             g                      Scalar              LIndF

Theoremindlcim 26476* 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 26477 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 26478* 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 15750 might be described as "every vector space is free." (Contributed by Stefan O'Rear, 26-Feb-2015.)
LBasis       Scalar       𝑚 freeLMod

16.15.48  Noetherian rings and left modules II

Syntaxclnr 26479 Extend class notation with the class of left Noetherian rings.
LNoeR

Definitiondf-lnr 26480 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 26481 Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeR ringLMod LNoeM

Theoremlnrrng 26482 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeR

Theoremlnrlnm 26483 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeR ringLMod LNoeM

Theoremislnr2 26484* Property of being a left-Noetherian ring in terms of finite generation of ideals (the usual "pure ring theory" definition). (Contributed by Stefan O'Rear, 24-Jan-2015.)
LIdeal       RSpan       LNoeR

Theoremislnr3 26485 Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       LNoeR NoeACS

Theoremlnr2i 26486* Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.)
LIdeal       RSpan       LNoeR

Theoremlpirlnr 26487 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LPIR LNoeR

Theoremlnrfrlm 26488 Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod        LNoeR LNoeM

Theoremlnrfg 26489 Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Scalar       LFinGen LNoeR LNoeM

Theoremlnrfgtr 26490 A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Scalar              s        LFinGen LNoeR LFinGen

16.15.49  Hilbert's Basis Theorem

Syntaxcldgis 26491 The leading ideal sequence used in the Hilbert Basis Theorem.
ldgIdlSeq

Definitiondf-ldgis 26492* Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree- elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 26500. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ldgIdlSeq LIdealPoly1 deg1 coe1

Theoremhbtlem1 26493* Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Poly1       LIdeal       ldgIdlSeq       deg1        coe1

Theoremhbtlem2 26494 Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq       LIdeal

Theoremhbtlem7 26495 Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq       LIdeal

Theoremhbtlem4 26496 The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq

Theoremhbtlem3 26497 The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Poly1       LIdeal       ldgIdlSeq

Theoremhbtlem5 26498* The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq

Theoremhbtlem6 26499* There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq       RSpan       LNoeR

Theoremhbt 26500 The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Poly1       LNoeR LNoeR

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