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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpl0 16201* The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmpladd 16202 The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly

Theoremmplmul 16203* The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                                                  g

Theoremmpl1 16204* The identity element of the ring of polynomials. (Contributed by Mario Carneiro, 10-Jan-2015.)
mPoly

Theoremmplsca 16205 The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                      Scalar

Theoremmplvsca2 16206 The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        mPwSer

Theoremmplvsca 16207* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly

Theoremmplvscaval 16208* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmvrcl 16209 A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        mVar

Theoremmplgrp 16210 The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmpllmod 16211 The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplrng 16212 The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplcrng 16213 The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplassa 16214 The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        AssAlg

Theoremressmplbas2 16215 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 16216 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmpladd 16217 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmplmul 16218 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmplvsca 16219 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremsubrgmpl 16220 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly               SubRing SubRing

Theoremsubrgmvr 16221 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 16222 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.)
mVar               SubRing       s        mPoly

Theoremmplmon 16223* A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplmonmul 16224* 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 16225* Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                                                                g

Theoremmplcoe3 16226* Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPoly                                    mulGrp       .g       mVar

Theoremmplcoe2 16227* 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 16228 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 16229* Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)
bag

Theoremltbwe 16230* 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 16231 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
ordPwSer

Theoremopsrval 16232* The value of the "ordered power series" function. This is the same as mPwSer psrval 16126, 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 16233* 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 16234 Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
mPwSer        ordPwSer                                    sSet

Theoremopsrbaslem 16235 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 16236 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 16237 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 16238 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 16239 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 16240 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 16241* Lemma for opsrtos 16243. (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer               Toset                     mPwSer                      bag

Theoremopsrtoslem2 16242* Lemma for opsrtos 16243. (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer               Toset                     mPwSer                      bag                             Toset

Theoremopsrtos 16243 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer               Toset                     Toset

Theoremopsrso 16244 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer               Toset

Theoremopsrcrng 16245 The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer

Theoremopsrassa 16246 The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
ordPwSer                             AssAlg

Theoremmplrcl 16247 Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPoly

Theoremmplelsfi 16248 A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.)
mPoly

Theoremmvrf2 16249 The power series/polynomial variable function maps indices to polynomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly        mVar

Theoremmplmon2 16250* Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theorempsrbag0 16251* The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theorempsrbagsn 16252* A singleton bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremmplascl 16253* Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                             algSc

Theoremmplasclf 16254 The scalar injection is a function into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                      algSc

Theoremsubrgascl 16255 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 16256 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 16257* A scaled monomial is a polynomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theoremmplmon2mul 16258* Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theoremmplind 16259* 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 16260* 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 16261* 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 16262* Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theorempsrbagev1 16263* A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
.g              CMnd

Theorempsrbagev2 16264* Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.)
.g              CMnd                            g

Theoremevlslem2 16265* 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 16266 Univariate power series.
PwSer1

Syntaxcv1 16267 The base variable of a univariate power series.
var1

Syntaxcpl1 16268 Univariate polynomials.
Poly1

Syntaxces1 16269 Evaluation in a subring.
evalSub1

Syntaxce1 16270 Evaluation of a univariate polynomial.
eval1

Syntaxcco1 16271 Convert a multivariate polynomial representation to univariate.
coe1

Syntaxctp1 16272 Convert a univariate polynomial representation to multivariate.
toPoly1

Definitiondf-psr1 16273 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 ordPwSer

Definitiondf-vr1 16274 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 16275 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 PwSer1s mPoly

Definitiondf-evls1 16276* 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 16277* 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 16278* Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Definitiondf-toply1 16279* 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 16280 The set of finite bags on is just the set of all functions from to . (Contributed by Mario Carneiro, 9-Feb-2015.)

Theorempsr1val 16281 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1       ordPwSer

Theorempsr1crng 16282 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1

Theorempsr1assa 16283 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1       AssAlg

Theorempsr1tos 16284 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
PwSer1       Toset Toset

Theorempsr1bas2 16285 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
PwSer1              mPwSer

Theorempsr1bas 16286 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1

Theoremvr1val 16287 The value of the generator of the power series algebra (the in ). Since all univariate polynomial rings over a fixed base ring are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
var1       mVar

Theoremvr1cl2 16288 The variable is a member of the power series algebra . (Contributed by Mario Carneiro, 8-Feb-2015.)
var1       PwSer1

Theoremply1val 16289 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1       s mPoly

Theoremply1bas 16290 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1              mPoly

Theoremply1lss 16291 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1

Theoremply1subrg 16292 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1              SubRing

Theoremply1crng 16293 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1

Theoremply1assa 16294 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       AssAlg

Theorempsr1rclOLD 16295 Obsolete version of elbasfv 13207 as of 5-Apr-2016. Reverse closure for ring existence from the univariate power series base set. (Contributed by Stefan O'Rear, 25-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
PwSer1

Theorempsr1bascl 16296 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1              mPwSer

Theorempsr1basf 16297 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1

Theoremply1rclOLD 16298 Obsolete version of elbasfv 13207 as of 5-Apr-2016. Reverse closure for ring existence from the univariate polynomial base set. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Poly1

Theoremply1basf 16299 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Poly1

Theoremply1bascl 16300 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1              PwSer1

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