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

Theoremmvrf 16001 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 16002 The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer        mVar

Theoremmvrcl2 16003 A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.)
mPwSer        mVar

Theoremreldmmpl 16004 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPoly

Theoremmplval 16005* 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 16006* 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 16007 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

Theoremmplval2 16008 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 16009 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 16010* An 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 16011* 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 16012* 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 16013 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 16014 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 16015* Lemma for mplsubrg 16016. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPwSer        mPoly

Theoremmplsubrg 16016 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 16017* The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmpladd 16018 The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly

Theoremmplmul 16019* The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                                                  g

Theoremmpl1 16020* The identity element of the ring of polynomials. (Contributed by Mario Carneiro, 10-Jan-2015.)
mPoly

Theoremmplsca 16021 The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                      Scalar

Theoremmplvsca2 16022 The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        mPwSer

Theoremmplvsca 16023* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly

Theoremmplvscaval 16024* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmvrcl 16025 A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        mVar

Theoremmplgrp 16026 The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmpllmod 16027 The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplrng 16028 The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplcrng 16029 The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplassa 16030 The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        AssAlg

Theoremressmplbas2 16031 The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       mPwSer

Theoremressmplbas 16032 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmpladd 16033 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmplmul 16034 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmplvsca 16035 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremsubrgmpl 16036 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly               SubRing SubRing

Theoremsubrgmvr 16037 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 4-Jul-2015.)
mVar               SubRing       s        mVar

Theoremsubrgmvrf 16038 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.)
mVar               SubRing       s        mPoly

Theoremmplmon 16039* A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplmonmul 16040* The product of two monomials adds the exponent vectors together. For example, the product of with is , where the exponent vectors and are added to give . (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplcoe1 16041* Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                                                                g

Theoremmplcoe3 16042* Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPoly                                    mulGrp       .g       mVar

Theoremmplcoe2 16043* Decompose a monomial into a finite product of powers of variables. (The assumption that is a commutative ring is not strictly necessary, because the submonoid of monomials is in the center of the multiplicative monoid of polynomials, but it simplifies the proof.) (Contributed by Mario Carneiro, 10-Jan-2015.)
mPoly                                    mulGrp       .g       mVar                      g

Theoremmplbas2 16044 An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 10-Jan-2015.)
mPoly        mPwSer        mVar        AlgSpan

Theoremltbval 16045* Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)
bag

Theoremltbwe 16046* The finite bag order is a well-order, given a well order of the index set. (Contributed by Mario Carneiro, 2-Jun-2015.)
bag

Theoremreldmopsr 16047 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
ordPwSer

Theoremopsrval 16048* The value of the "ordered power series" function. This is the same as mPwSer psrval 15942, but with the addition of a well-order on we can turn a strict order on into a strict order on the power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
mPwSer        ordPwSer                      bag                                           sSet

Theoremopsrle 16049* An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer        ordPwSer                      bag

Theoremopsrval2 16050 Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
mPwSer        ordPwSer                                    sSet

Theoremopsrbaslem 16051 Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer        ordPwSer               Slot

Theoremopsrbas 16052 The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPwSer        ordPwSer

Theoremopsrplusg 16053 The addition operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPwSer        ordPwSer

Theoremopsrmulr 16054 The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPwSer        ordPwSer

Theoremopsrvsca 16055 The scalar product operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPwSer        ordPwSer

Theoremopsrsca 16056 The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPwSer        ordPwSer                             Scalar

Theoremopsrtoslem1 16057* Lemma for opsrtos 16059. (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer               Toset                     mPwSer                      bag

Theoremopsrtoslem2 16058* Lemma for opsrtos 16059. (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer               Toset                     mPwSer                      bag                             Toset

Theoremopsrtos 16059 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer               Toset                     Toset

Theoremopsrso 16060 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer               Toset

Theoremopsrcrng 16061 The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer

Theoremopsrassa 16062 The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
ordPwSer                             AssAlg

Theoremmplrcl 16063 Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPoly

Theoremmplelsfi 16064 A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.)
mPoly

Theoremmvrf2 16065 The power series/polynomial variable function maps indices to polynomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly        mVar

Theoremmplmon2 16066* Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theorempsrbag0 16067* The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theorempsrbagsn 16068* A singleton bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremmplascl 16069* Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                             algSc

Theoremmplasclf 16070 The scalar injection is a function into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                      algSc

Theoremsubrgascl 16071 The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
mPoly        algSc       s        mPoly               SubRing       algSc

Theoremsubrgasclcl 16072 The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
mPoly        algSc       s        mPoly               SubRing

Theoremmplmon2cl 16073* A scaled monomial is a polynomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theoremmplmon2mul 16074* Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theoremmplind 16075* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. The commutativity condition is stronger than strictly needed. (Contributed by Stefan O'Rear, 11-Mar-2015.)
mVar        mPoly                      algSc

Theoremmplcoe4 16076* Decompose a polynomial into a finite sum of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly                                                  g

10.10.2  Polynomial evaluation

Theoremevlslem4 16077* The support of a tensor product of ring element families is contained in the product of the supports. (Contributed by Stefan O'Rear, 8-Mar-2015.)

Theorempsrbagsuppfi 16078* Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theorempsrbagev1 16079* A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
.g              CMnd

Theorempsrbagev2 16080* Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.)
.g              CMnd                            g

Theoremevlslem2 16081* A linear function on the polynomial ring which is multiplicative on scaled monomials is generally multiplicative. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly

10.10.3  Univariate Polynomials

Syntaxcps1 16082 Univariate power series.
PwSer1

Syntaxcv1 16083 The base variable of a univariate power series.
var1

Syntaxcpl1 16084 Univariate polynomials.
Poly1

Syntaxces1 16085 Evaluation in a subring.
evalSub1

Syntaxce1 16086 Evaluation of a univariate polynomial.
eval1

Syntaxcco1 16087 Convert a multivariate polynomial representation to univariate.
coe1

Syntaxctp1 16088 Convert a univariate polynomial representation to multivariate.
toPoly1

Definitiondf-psr1 16089 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 ordPwSer

Definitiondf-vr1 16090 Define the base element of a univariate power series (the element of the set of polynomials and also the in the set of power series). (Contributed by Mario Carneiro, 8-Feb-2015.)
var1 mVar

Definitiondf-ply1 16091 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 PwSer1s mPoly

Definitiondf-evls1 16092* Define the evaluation map for the univariate polynomial algebra. The function evalSub1 makes sense when is a ring and is a subring of , and where is the set of polynomials in Poly1. This function maps an element of the formal polynomial algebra (with coefficients in ) to a function from assignments to the variable from into an element of formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
evalSub1 evalSub

Definitiondf-evl1 16093* Define the evaluation map for the univariate polynomial algebra. The function eval1 makes sense when is a ring, and is the set of polynomials in Poly1. This function maps an element of the formal polynomial algebra (with coefficients in ) to a function from assignments to the variable from into an element of formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1 eval

Definitiondf-coe1 16094* Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Definitiondf-toply1 16095* Define a function which maps a coefficient function for a univariate polynomial to the corresponding polynomial object. (Contributed by Mario Carneiro, 12-Jun-2015.)
toPoly1

Theorempsr1baslem 16096 The set of finite bags on is just the set of all functions from to . (Contributed by Mario Carneiro, 9-Feb-2015.)

Theorempsr1val 16097 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1       ordPwSer

Theorempsr1crng 16098 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1

Theorempsr1assa 16099 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1       AssAlg

Theorempsr1tos 16100 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
PwSer1       Toset Toset

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