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Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-assa 16001* Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) ring, coupled with an multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.)
AssAlg Scalar

Definitiondf-asp 16002* Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan AssAlg SubRing

Definitiondf-ascl 16003* Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unit. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc Scalar

Theoremisassa 16004* The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassalem 16005 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassaass 16006 Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassaassr 16007 Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Scalar                            AssAlg

Theoremassalmod 16008 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
AssAlg

Theoremassarng 16009 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
AssAlg

Theoremassasca 16010 An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Scalar       AssAlg

Theoremisassad 16011* Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
Scalar                                                               AssAlg

Theoremissubassa 16012 The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
s                             AssAlg AssAlg SubRing

Theoremsraassa 16013 The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
subringAlg        SubRing AssAlg

Theoremrlmassa 16014 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod AssAlg

Theoremassapropd 16015* If two structures have the same components (properties), one is an associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
Scalar       Scalar                     AssAlg AssAlg

Theoremaspval 16016* Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg SubRing

Theoremasplss 16017 The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg

Theoremaspid 16018 The algebraic span of a subalgebra is itself. (spanid 21872 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg SubRing

Theoremaspsubrg 16019 The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg SubRing

Theoremaspss 16020 Span preserves subset ordering. (spanss 21873 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg

Theoremaspssid 16021 A set of vectors is a subset of its span. (spanss2 21870 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg

Theoremasclfval 16022* Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar

Theoremasclval 16023 Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar

Theoremasclfn 16024 Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar

Theoremasclf 16025 The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
algSc       Scalar

Theoremasclghm 16026 The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
algSc       Scalar

Theoremasclmul1 16027 Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar                                   AssAlg

Theoremasclmul2 16028 Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar                                   AssAlg

Theoremasclrhm 16029 The scalar injection is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar       AssAlg RingHom

Theoremrnascl 16030 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc                     AssAlg

Theoremressascl 16031 The injection of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       s        SubRing algSc

Theoremissubassa2 16032 A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc              AssAlg SubRing

Theoremasclpropd 16033* If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on can be discharged either by letting (if strong equality is known on ) or assuming is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
Scalar       Scalar                                          algSc algSc

Theoremaspval2 16034 The algebraic closure is the ring closure when the generating set is expanded to include all scalars. EDITORIAL : In light of this, is AlgSpan independently needed? (Contributed by Stefan O'Rear, 9-Mar-2015.)
AlgSpan       algSc       mrClsSubRing              AssAlg

10.10  Abstract Multivariate Polynomials

10.10.1  Definition and basic properties

Syntaxcmps 16035 Multivariate power series.
mPwSer

Syntaxcmvr 16036 Multivariate power series variables.
mVar

Syntaxcmpl 16037 Multivariate polynomials.
mPoly

Syntaxces 16038 Evaluation in a superring.
evalSub

Syntaxcevl 16039 Evaluation of a multivariate polynomial.
eval

Syntaxcmhp 16040 Multivariate polynomials.
mHomP

Syntaxcpsd 16041 Power series partial derivative function.
mPSDer

Syntaxcltb 16042 Ordering on terms of a multivariate polynomial.
bag

Syntaxcopws 16043 Ordered set of power series.
ordPwSer

Syntaxcslv 16044 Select a subset of variables in a multivariate polynomial.
selectVars

Syntaxcai 16045 Algebraically independent.
AlgInd

Definitiondf-psr 16046* Define the algebra of power series over the index set and with coefficients from the ring . (Contributed by Mario Carneiro, 21-Mar-2015.)
mPwSer g Scalar TopSet

Definitiondf-mvr 16047* Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Definitiondf-mpl 16048* Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). (Contributed by Mario Carneiro, 7-Jan-2015.)
mPoly mPwSer s

Definitiondf-evls 16049* Define the evaluation map for the polynomial algebra. The function evalSub makes sense when is an index set, is a ring, is a subring of , and where is the set of polynomials in mPoly . This function maps an element of the formal polynomial algebra (with coefficients in ) to a function from assignments of the variables to elements of formed by evaluating the polynomial with the given assignments. (Contributed by Stefan O'Rear, 11-Mar-2015.)
evalSub SubRing mPoly s RingHom s algSc mVar s

Definitiondf-evl 16050* A simplication of evalSub when the evaluation ring is the same as the coefficient ring. (Contributed by Stefan O'Rear, 19-Mar-2015.)
eval evalSub

Definitiondf-mhp 16051* Define the subspaces of order- homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mHomP mPoly

Definitiondf-psd 16052* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPSDer mPwSer .g

Definitiondf-ltbag 16053* Define a well-order on the set of all finite bags from the index set given a wellordering of . (Contributed by Mario Carneiro, 8-Feb-2015.)
bag

Definitiondf-opsr 16054* Define a total order on the set of all power series in from the index set given a wellordering of and a totally ordered base ring . (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer mPwSer sSet bag

Definitiondf-selv 16055* Define the "variable selection" function. The function selectVars maps elements of mPoly bijectively onto mPoly mPoly in the natural way, for example if and it would map mPoly to mPoly mPoly . This, for example, allows one to treat a multivariate polynomial as a univariate polynomial with coefficients in a polynomial ring with one less variable. (Contributed by Mario Carneiro, 21-Mar-2015.)
selectVars mPoly mPoly Scalar evalSub s mVar mPoly mVar

Definitiondf-algind 16056* Define the predicate "the set is algebraically independent in the algebra ". A collection of vectors is algebraically independent if no nontrivial polynomial with elements from the subset evaluates to zero. (Contributed by Mario Carneiro, 21-Mar-2015.)
AlgInd mPoly s evalSub

Theoremreldmpsr 16057 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPwSer

Theorempsrval 16058* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer                                                         g                                    Scalar TopSet

Theorempsrvalstr 16059 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
Scalar TopSet Struct

Theorempsrbag 16060* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagf 16061* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglesupp 16062* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglecl 16063* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagaddcl 16064* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)

Theorempsrbagcon 16065* The analogue of the statement " implies " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglefi 16066* There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.)

Theorempsrbagconcl 16067* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagconf1o 16068* Bag complementation is a bijection on the set of bags dominated by a given bag . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremgsumbagdiaglem 16069* Lemma for gsumbagdiag 16070. (Contributed by Mario Carneiro, 5-Jan-2015.)

Theoremgsumbagdiag 16070* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12191 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd              g g

Theorempsrass1lem 16071* A group sum commutation used by psrass1 16098. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd                     g g g g

Theorempsrbas 16072* The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer

Theorempsrelbas 16073* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrplusg 16074 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer

Theorempsradd 16075 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsraddcl 16076 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrmulr 16077* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer                                    g

Theorempsrmulfval 16078* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                  g

Theorempsrmulval 16079* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                         g

Theorempsrmulcllem 16080* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrmulcl 16081 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrsca 16082 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                      Scalar

Theorempsrvscafval 16083* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer

Theorempsrvsca 16084* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscaval 16085* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscacl 16086 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0cl 16087* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0lid 16088* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrnegcl 16089* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlinv 16090* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrgrp 16091 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0 16092* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrneg 16093* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlmod 16094 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr1cl 16095* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlidm 16096* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrridm 16097* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrass1 16098* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
mPwSer

Theorempsrdi 16099* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrdir 16100* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

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