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Theorem List for Metamath Proof Explorer - 16501-16600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpllss 16501 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 16502* Lemma for mplsubrg 16503. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPwSer        mPoly

Theoremmplsubrg 16503 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 16504* The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmpladd 16505 The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly

Theoremmplmul 16506* The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                                                  g

Theoremmpl1 16507* The identity element of the ring of polynomials. (Contributed by Mario Carneiro, 10-Jan-2015.)
mPoly

Theoremmplsca 16508 The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                      Scalar

Theoremmplvsca2 16509 The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        mPwSer

Theoremmplvsca 16510* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly

Theoremmplvscaval 16511* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmvrcl 16512 A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        mVar

Theoremmplgrp 16513 The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmpllmod 16514 The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplrng 16515 The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplcrng 16516 The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplassa 16517 The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly        AssAlg

Theoremressmplbas2 16518 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 16519 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmpladd 16520 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmplmul 16521 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremressmplvsca 16522 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly                      SubRing       s

Theoremsubrgmpl 16523 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
mPoly        s        mPoly               SubRing SubRing

Theoremsubrgmvr 16524 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 16525 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.)
mVar               SubRing       s        mPoly

Theoremmplmon 16526* A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly

Theoremmplmonmul 16527* 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 16528* Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
mPoly                                                                g

Theoremmplcoe3 16529* Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.)
mPoly                                    mulGrp       .g       mVar

Theoremmplcoe2 16530* 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 16531 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 16532* Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)
bag

Theoremltbwe 16533* 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 16534 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
ordPwSer

Theoremopsrval 16535* The value of the "ordered power series" function. This is the same as mPwSer psrval 16429, 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 16536* 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 16537 Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
mPwSer        ordPwSer                                    sSet

Theoremopsrbaslem 16538 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 16539 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 16540 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 16541 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 16542 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 16543 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 16544* Lemma for opsrtos 16546. (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer               Toset                     mPwSer                      bag

Theoremopsrtoslem2 16545* Lemma for opsrtos 16546. (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer               Toset                     mPwSer                      bag                             Toset

Theoremopsrtos 16546 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer               Toset                     Toset

Theoremopsrso 16547 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer               Toset

Theoremopsrcrng 16548 The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer

Theoremopsrassa 16549 The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
ordPwSer                             AssAlg

Theoremmplrcl 16550 Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPoly

Theoremmplelsfi 16551 A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.)
mPoly

Theoremmvrf2 16552 The power series/polynomial variable function maps indices to polynomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly        mVar

Theoremmplmon2 16553* Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theorempsrbag0 16554* The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theorempsrbagsn 16555* A singleton bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremmplascl 16556* Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                             algSc

Theoremmplasclf 16557 The scalar injection is a function into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                      algSc

Theoremsubrgascl 16558 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 16559 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 16560* A scaled monomial is a polynomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theoremmplmon2mul 16561* Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theoremmplind 16562* 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 16563* 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 16564* 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 16565* Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theorempsrbagev1 16566* A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
.g              CMnd

Theorempsrbagev2 16567* Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.)
.g              CMnd                            g

Theoremevlslem2 16568* 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 16569 Univariate power series.
PwSer1

Syntaxcv1 16570 The base variable of a univariate power series.
var1

Syntaxcpl1 16571 Univariate polynomials.
Poly1

Syntaxces1 16572 Evaluation in a subring.
evalSub1

Syntaxce1 16573 Evaluation of a univariate polynomial.
eval1

Syntaxcco1 16574 Convert a multivariate polynomial representation to univariate.
coe1

Syntaxctp1 16575 Convert a univariate polynomial representation to multivariate.
toPoly1

Definitiondf-psr1 16576 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 ordPwSer

Definitiondf-vr1 16577 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 16578 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 PwSer1s mPoly

Definitiondf-evls1 16579* 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 16580* 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 16581* Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Definitiondf-toply1 16582* 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 16583 The set of finite bags on is just the set of all functions from to . (Contributed by Mario Carneiro, 9-Feb-2015.)

Theorempsr1val 16584 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1       ordPwSer

Theorempsr1crng 16585 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1

Theorempsr1assa 16586 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1       AssAlg

Theorempsr1tos 16587 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
PwSer1       Toset Toset

Theorempsr1bas2 16588 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
PwSer1              mPwSer

Theorempsr1bas 16589 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1

Theoremvr1val 16590 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 16591 The variable is a member of the power series algebra . (Contributed by Mario Carneiro, 8-Feb-2015.)
var1       PwSer1

Theoremply1val 16592 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1       s mPoly

Theoremply1bas 16593 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1              mPoly

Theoremply1lss 16594 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1

Theoremply1subrg 16595 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1              SubRing

Theoremply1crng 16596 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1

Theoremply1assa 16597 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       AssAlg

Theorempsr1bascl 16598 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1              mPwSer

Theorempsr1basf 16599 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1

Theoremply1basf 16600 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Poly1

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