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

Theoremopsrtos 16501 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer               Toset                     Toset

Theoremopsrso 16502 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer               Toset

Theoremopsrcrng 16503 The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
ordPwSer

Theoremopsrassa 16504 The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
ordPwSer                             AssAlg

Theoremmplrcl 16505 Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
mPoly

Theoremmplelsfi 16506 A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.)
mPoly

Theoremmvrf2 16507 The power series/polynomial variable function maps indices to polynomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly        mVar

Theoremmplmon2 16508* Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theorempsrbag0 16509* The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theorempsrbagsn 16510* A singleton bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremmplascl 16511* Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                             algSc

Theoremmplasclf 16512 The scalar injection is a function into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                      algSc

Theoremsubrgascl 16513 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 16514 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 16515* A scaled monomial is a polynomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theoremmplmon2mul 16516* Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly

Theoremmplind 16517* 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 16518* 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 16519* 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 16520* Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theorempsrbagev1 16521* A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
.g              CMnd

Theorempsrbagev2 16522* Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.)
.g              CMnd                            g

Theoremevlslem2 16523* 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 16524 Univariate power series.
PwSer1

Syntaxcv1 16525 The base variable of a univariate power series.
var1

Syntaxcpl1 16526 Univariate polynomials.
Poly1

Syntaxces1 16527 Evaluation in a subring.
evalSub1

Syntaxce1 16528 Evaluation of a univariate polynomial.
eval1

Syntaxcco1 16529 Convert a multivariate polynomial representation to univariate.
coe1

Syntaxctp1 16530 Convert a univariate polynomial representation to multivariate.
toPoly1

Definitiondf-psr1 16531 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 ordPwSer

Definitiondf-vr1 16532 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 16533 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 PwSer1s mPoly

Definitiondf-evls1 16534* 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 16535* 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 16536* Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Definitiondf-toply1 16537* 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 16538 The set of finite bags on is just the set of all functions from to . (Contributed by Mario Carneiro, 9-Feb-2015.)

Theorempsr1val 16539 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1       ordPwSer

Theorempsr1crng 16540 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1

Theorempsr1assa 16541 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1       AssAlg

Theorempsr1tos 16542 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
PwSer1       Toset Toset

Theorempsr1bas2 16543 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
PwSer1              mPwSer

Theorempsr1bas 16544 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1

Theoremvr1val 16545 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 16546 The variable is a member of the power series algebra . (Contributed by Mario Carneiro, 8-Feb-2015.)
var1       PwSer1

Theoremply1val 16547 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1       s mPoly

Theoremply1bas 16548 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1              mPoly

Theoremply1lss 16549 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1

Theoremply1subrg 16550 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1              SubRing

Theoremply1crng 16551 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1

Theoremply1assa 16552 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       AssAlg

Theorempsr1bascl 16553 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1              mPwSer

Theorempsr1basf 16554 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1

Theoremply1basf 16555 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Poly1

Theoremply1bascl 16556 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1              PwSer1

Theoremply1bascl2 16557 A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1              mPoly

Theoremcoe1fval 16558* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremcoe1fv 16559 Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremfvcoe1 16560 Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremcoe1fval3 16561* Univariate power series coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
coe1              PwSer1

Theoremcoe1f2 16562 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
coe1              PwSer1

Theoremcoe1fval2 16563* Univariate polynomial coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremcoe1f 16564 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremcoe1sfi 16565 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremvr1cl 16566 The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
var1       Poly1

Theoremopsr0 16567 Zero in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer        ordPwSer

Theoremopsr1 16568 One in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer        ordPwSer

Theoremmplplusg 16569 Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

Theoremmplmulr 16570 Value of multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

Theorempsr1plusg 16571 Value of addition in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
PwSer1       mPwSer

Theorempsr1vsca 16572 Value of scalar multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
PwSer1       mPwSer

Theorempsr1mulr 16573 Value of multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
PwSer1       mPwSer

Theoremply1plusg 16574 Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
Poly1       mPoly

Theoremply1vsca 16575 Value of scalar multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
Poly1       mPoly

Theoremply1mulr 16576 Value of multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
Poly1       mPoly

Theoremressply1bas2 16577 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.)
Poly1       s        Poly1              SubRing       PwSer1

Theoremressply1bas 16578 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1add 16579 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1mul 16580 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1vsca 16581 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremsubrgply1 16582 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing SubRing

Theorempsrbaspropd 16583 Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPwSer mPwSer

Theorempsrplusgpropd 16584* Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
mPwSer mPwSer

Theoremmplbaspropd 16585* Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPoly mPoly

Theoremstrov2rcl 16586 Reverse closure for polynomial-resembling things. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Theorempsropprmul 16587 Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPwSer        oppr       mPwSer

Theoremply1opprmul 16588 Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       oppr       Poly1

Theorem00ply1bas 16589 Lemma for ply1basfvi 16590 and deg1fvi 19961. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1

Theoremply1basfvi 16590 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1plusgfvi 16591 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1baspropd 16592* Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1plusgpropd 16593* Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremopsrrng 16594 Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
ordPwSer

Theoremopsrlmod 16595 Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
ordPwSer

Theorempsr1rng 16596 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
PwSer1

Theoremply1rng 16597 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Poly1

Theorempsr1lmod 16598 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
PwSer1

Theorempsr1sca 16599 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
PwSer1       Scalar

Theorempsr1sca2 16600 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
PwSer1       Scalar

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