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Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrnascl 16401 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc                     AssAlg

Theoremressascl 16402 The injection of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)
algSc       s        SubRing algSc

Theoremissubassa2 16403 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 16404* 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 16405 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 16406 Multivariate power series.
mPwSer

Syntaxcmvr 16407 Multivariate power series variables.
mVar

Syntaxcmpl 16408 Multivariate polynomials.
mPoly

Syntaxces 16409 Evaluation in a superring.
evalSub

Syntaxcevl 16410 Evaluation of a multivariate polynomial.
eval

Syntaxcmhp 16411 Multivariate polynomials.
mHomP

Syntaxcpsd 16412 Power series partial derivative function.
mPSDer

Syntaxcltb 16413 Ordering on terms of a multivariate polynomial.
bag

Syntaxcopws 16414 Ordered set of power series.
ordPwSer

Syntaxcslv 16415 Select a subset of variables in a multivariate polynomial.
selectVars

Syntaxcai 16416 Algebraically independent.
AlgInd

Definitiondf-psr 16417* 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 16418* Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Definitiondf-mpl 16419* 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 16420* 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 16421* 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 16422* Define the subspaces of order- homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mHomP mPoly

Definitiondf-psd 16423* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPSDer mPwSer .g

Definitiondf-ltbag 16424* 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 16425* 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 16426* 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 16427* 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 16428 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPwSer

Theorempsrval 16429* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer                                                         g                                    Scalar TopSet

Theorempsrvalstr 16430 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
Scalar TopSet Struct

Theorempsrbag 16431* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagf 16432* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglesupp 16433* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglecl 16434* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagaddcl 16435* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)

Theorempsrbagcon 16436* The analogue of the statement " implies " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglefi 16437* 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 16438* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagconf1o 16439* Bag complementation is a bijection on the set of bags dominated by a given bag . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremgsumbagdiaglem 16440* Lemma for gsumbagdiag 16441. (Contributed by Mario Carneiro, 5-Jan-2015.)

Theoremgsumbagdiag 16441* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12561 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd              g g

Theorempsrass1lem 16442* A group sum commutation used by psrass1 16469. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd                     g g g g

Theorempsrbas 16443* 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 16444* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrplusg 16445 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 16446 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsraddcl 16447 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrmulr 16448* 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 16449* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                  g

Theorempsrmulval 16450* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                         g

Theorempsrmulcllem 16451* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrmulcl 16452 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrsca 16453 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                      Scalar

Theorempsrvscafval 16454* 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 16455* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscaval 16456* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscacl 16457 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0cl 16458* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0lid 16459* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrnegcl 16460* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlinv 16461* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrgrp 16462 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0 16463* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrneg 16464* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlmod 16465 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr1cl 16466* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlidm 16467* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrridm 16468* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrass1 16469* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
mPwSer

Theorempsrdi 16470* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrdir 16471* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrcom 16472* Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrass23 16473* Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrrng 16474 The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr1 16475* The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
mPwSer

Theorempsrcrng 16476 The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
mPwSer

Theorempsrassa 16477 The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer                      AssAlg

Theoremresspsrbas 16478 A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsradd 16479 A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsrmul 16480 A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsrvsca 16481 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremsubrgpsr 16482 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 16483* A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremmvrfval 16484* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrval 16485* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrval2 16486* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrid 16487* The -th coefficient of the term is . (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrf 16488 The power series variable function is a function from the index set to elements of the power series structure representing for each . (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer        mVar

Theoremmvrf1 16489 The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer        mVar

Theoremmvrcl2 16490 A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer        mVar

Theoremreldmmpl 16491 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPoly

Theoremmplval 16492* Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer                             s

Theoremmplbas 16493* Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

Theoremmplelbas 16494 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

Theoremmplval2 16495 Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer               s

Theoremmplbasss 16496 The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

Theoremmplelf 16497* A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly

Theoremmplsubglem 16498* If is an ideal of sets (a nonempty collection closed under subset and binary union) of the set of finite bags (the primary applications being and for some ), then the set of all power series whose coefficient functions are supported on an element of is a subgroup of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
mPwSer                                                                       SubGrp

Theoremmpllsslem 16499* If is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set of finite bags (the primary applications being and for some ), then the set of all power series whose coefficient functions are supported on an element of is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
mPwSer

Theoremmplsubg 16500 The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
mPwSer        mPoly                             SubGrp

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