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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremasplss 15901 The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg

Theoremaspid 15902 The algebraic span of a subalgebra is itself. (spanid 21756 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan                     AssAlg SubRing

Theoremaspsubrg 15903 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 15904 Span preserves subset ordering. (spanss 21757 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg

Theoremaspssid 15905 A set of vectors is a subset of its span. (spanss2 21754 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan              AssAlg

Theoremasclfval 15906* Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar

Theoremasclval 15907 Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar

Theoremasclfn 15908 Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar

Theoremasclf 15909 The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
algSc       Scalar

Theoremasclghm 15910 The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
algSc       Scalar

Theoremasclmul1 15911 Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar                                   AssAlg

Theoremasclmul2 15912 Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       Scalar                                   AssAlg

Theoremasclrhm 15913 The scalar injection is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc       Scalar       AssAlg RingHom

Theoremrnascl 15914 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc                     AssAlg

Theoremressascl 15915 The injection of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       s        SubRing algSc

Theoremissubassa2 15916 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 15917* 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 15918 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 15919 Multivariate power series.
mPwSer

Syntaxcmvr 15920 Multivariate power series variables.
mVar

Syntaxcmpl 15921 Multivariate polynomials.
mPoly

Syntaxces 15922 Evaluation in a superring.
evalSub

Syntaxcevl 15923 Evaluation of a multivariate polynomial.
eval

Syntaxcmhp 15924 Multivariate polynomials.
mHomP

Syntaxcpsd 15925 Power series partial derivative function.
mPSDer

Syntaxcltb 15926 Ordering on terms of a multivariate polynomial.
bag

Syntaxcopws 15927 Ordered set of power series.
ordPwSer

Syntaxcslv 15928 Select a subset of variables in a multivariate polynomial.
selectVars

Syntaxcai 15929 Algebraically independent.
AlgInd

Definitiondf-psr 15930* 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 15931* Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Definitiondf-mpl 15932* 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 15933* 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 15934* 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 15935* Define the subspaces of order- homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mHomP mPoly

Definitiondf-psd 15936* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPSDer mPwSer .g

Definitiondf-ltbag 15937* 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 15938* 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 15939* 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 15940* 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 15941 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPwSer

Theorempsrval 15942* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer                                                         g                                    Scalar TopSet

Theorempsrvalstr 15943 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
Scalar TopSet Struct

Theorempsrbag 15944* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagf 15945* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglesupp 15946* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglecl 15947* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagaddcl 15948* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)

Theorempsrbagcon 15949* The analogue of the statement " implies " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglefi 15950* 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 15951* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagconf1o 15952* Bag complementation is a bijection on the set of bags dominated by a given bag . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremgsumbagdiaglem 15953* Lemma for gsumbagdiag 15954. (Contributed by Mario Carneiro, 5-Jan-2015.)

Theoremgsumbagdiag 15954* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12117 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd              g g

Theorempsrass1lem 15955* A group sum commutation used by psrass1 15982. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd                     g g g g

Theorempsrbas 15956* 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 15957* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrplusg 15958 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 15959 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsraddcl 15960 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrmulr 15961* 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 15962* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                  g

Theorempsrmulval 15963* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                         g

Theorempsrmulcllem 15964* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrmulcl 15965 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrsca 15966 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                      Scalar

Theorempsrvscafval 15967* 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 15968* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscaval 15969* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscacl 15970 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0cl 15971* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0lid 15972* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrnegcl 15973* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlinv 15974* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrgrp 15975 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0 15976* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrneg 15977* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlmod 15978 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr1cl 15979* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlidm 15980* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrridm 15981* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrass1 15982* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
mPwSer

Theorempsrdi 15983* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrdir 15984* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrcom 15985* Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrass23 15986* Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrrng 15987 The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr1 15988* The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
mPwSer

Theorempsrcrng 15989 The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
mPwSer

Theorempsrassa 15990 The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer                      AssAlg

Theoremresspsrbas 15991 A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsradd 15992 A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsrmul 15993 A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsrvsca 15994 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremsubrgpsr 15995 A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               SubRing SubRing

Theoremmvridlem 15996* A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremmvrfval 15997* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrval 15998* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrval2 15999* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrid 16000* The -th coefficient of the term is . (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

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