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Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcevl 16301 Evaluation of a multivariate polynomial.
eval

Syntaxcmhp 16302 Multivariate polynomials.
mHomP

Syntaxcpsd 16303 Power series partial derivative function.
mPSDer

Syntaxcltb 16304 Ordering on terms of a multivariate polynomial.
bag

Syntaxcopws 16305 Ordered set of power series.
ordPwSer

Syntaxcslv 16306 Select a subset of variables in a multivariate polynomial.
selectVars

Syntaxcai 16307 Algebraically independent.
AlgInd

Definitiondf-psr 16308* 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 16309* Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Definitiondf-mpl 16310* 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 16311* 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 16312* 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 16313* Define the subspaces of order- homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mHomP mPoly

Definitiondf-psd 16314* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPSDer mPwSer .g

Definitiondf-ltbag 16315* 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 16316* 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 16317* 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 16318* 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 16319 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPwSer

Theorempsrval 16320* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer                                                         g                                    Scalar TopSet

Theorempsrvalstr 16321 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
Scalar TopSet Struct

Theorempsrbag 16322* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagf 16323* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglesupp 16324* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglecl 16325* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagaddcl 16326* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)

Theorempsrbagcon 16327* The analogue of the statement " implies " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbaglefi 16328* 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 16329* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theorempsrbagconf1o 16330* Bag complementation is a bijection on the set of bags dominated by a given bag . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremgsumbagdiaglem 16331* Lemma for gsumbagdiag 16332. (Contributed by Mario Carneiro, 5-Jan-2015.)

Theoremgsumbagdiag 16332* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12448 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd              g g

Theorempsrass1lem 16333* A group sum commutation used by psrass1 16360. (Contributed by Mario Carneiro, 5-Jan-2015.)
CMnd                     g g g g

Theorempsrbas 16334* 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 16335* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrplusg 16336 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 16337 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsraddcl 16338 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrmulr 16339* 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 16340* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                  g

Theorempsrmulval 16341* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                                                         g

Theorempsrmulcllem 16342* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrmulcl 16343 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrsca 16344 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer                      Scalar

Theorempsrvscafval 16345* 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 16346* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscaval 16347* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
mPwSer

Theorempsrvscacl 16348 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0cl 16349* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0lid 16350* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrnegcl 16351* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlinv 16352* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrgrp 16353 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr0 16354* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrneg 16355* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlmod 16356 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr1cl 16357* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrlidm 16358* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrridm 16359* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsrass1 16360* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
mPwSer

Theorempsrdi 16361* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrdir 16362* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrcom 16363* Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrass23 16364* Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer

Theorempsrrng 16365 The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer

Theorempsr1 16366* The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
mPwSer

Theorempsrcrng 16367 The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
mPwSer

Theorempsrassa 16368 The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer                      AssAlg

Theoremresspsrbas 16369 A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsradd 16370 A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsrmul 16371 A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremresspsrvsca 16372 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPwSer        s        mPwSer               s        SubRing

Theoremsubrgpsr 16373 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 16374* A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremmvrfval 16375* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrval 16376* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrval2 16377* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrid 16378* The -th coefficient of the term is . (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar

Theoremmvrf 16379 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 16380 The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer        mVar

Theoremmvrcl2 16381 A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer        mVar

Theoremreldmmpl 16382 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPoly

Theoremmplval 16383* 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 16384* 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 16385 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

Theoremmplval2 16386 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 16387 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 16388* 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 16389* 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 16390* 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 16391 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

Theoremmpllss 16392 The set of polynomials is closed under scalar multiplication, i.e. it is a linear subspace of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPwSer        mPoly

Theoremmplsubrglem 16393* Lemma for mplsubrg 16394. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPwSer        mPoly

Theoremmplsubrg 16394 The set of polynomials is closed under multiplication, i.e. it is a subring of the set of power series. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPwSer        mPoly                             SubRing

Theoremmpl0 16395* The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmpladd 16396 The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly

Theoremmplmul 16397* The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                                                  g

Theoremmpl1 16398* The identity element of the ring of polynomials. (Contributed by Mario Carneiro, 10-Jan-2015.)
mPoly

Theoremmplsca 16399 The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                      Scalar

Theoremmplvsca2 16400 The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        mPwSer

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