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Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremply1bascl 16601 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1              PwSer1

Theoremply1bascl2 16602 A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1              mPoly

Theoremcoe1fval 16603* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremcoe1fv 16604 Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremfvcoe1 16605 Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremcoe1fval3 16606* Univariate power series coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
coe1              PwSer1

Theoremcoe1f2 16607 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
coe1              PwSer1

Theoremcoe1fval2 16608* Univariate polynomial coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremcoe1f 16609 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremcoe1sfi 16610 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremvr1cl 16611 The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
var1       Poly1

Theoremopsr0 16612 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 16613 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 16614 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 16615 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 16616 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 16617 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 16618 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 16619 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 16620 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 16621 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 16622 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 16623 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1add 16624 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1mul 16625 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1vsca 16626 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremsubrgply1 16627 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing SubRing

Theorempsrbaspropd 16628 Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPwSer mPwSer

Theorempsrplusgpropd 16629* Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
mPwSer mPwSer

Theoremmplbaspropd 16630* Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPoly mPoly

Theoremstrov2rcl 16631 Reverse closure for polynomial-resembling things. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Theorempsropprmul 16632 Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPwSer        oppr       mPwSer

Theoremply1opprmul 16633 Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       oppr       Poly1

Theorem00ply1bas 16634 Lemma for ply1basfvi 16635 and deg1fvi 20008. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1

Theoremply1basfvi 16635 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1plusgfvi 16636 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1baspropd 16637* Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1plusgpropd 16638* Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremopsrrng 16639 Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
ordPwSer

Theoremopsrlmod 16640 Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
ordPwSer

Theorempsr1rng 16641 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
PwSer1

Theoremply1rng 16642 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Poly1

Theorempsr1lmod 16643 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
PwSer1

Theorempsr1sca 16644 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
PwSer1       Scalar

Theorempsr1sca2 16645 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
PwSer1       Scalar

Theoremply1lmod 16646 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1

Theoremply1sca 16647 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1       Scalar

Theoremply1sca2 16648 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1       Scalar

Theoremply1mpl0 16649 The univariate polynomial ring has the same zero as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
mPoly        Poly1

Theoremply1mpl1 16650 The univariate polynomial ring has the same one as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
mPoly        Poly1

Theoremply1ascl 16651 The univariate polynomial ring inherits the multivariate ring's scalar function. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by Fan Zheng, 26-Jun-2016.)
Poly1       algSc       algSc mPoly

Theoremsubrg1ascl 16652 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.)
Poly1       algSc       s        Poly1       SubRing       algSc

Theoremsubrg1asclcl 16653 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.)
Poly1       algSc       s        Poly1       SubRing

Theoremsubrgvr1 16654 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
var1       SubRing       s        var1

Theoremsubrgvr1cl 16655 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
var1       SubRing       s        Poly1

Theoremcoe1z 16656 The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Poly1                     coe1

Theoremcoe1add 16657 The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
Poly1                            coe1 coe1 coe1

Theoremcoe1addfv 16658 A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Poly1                            coe1 coe1 coe1

Theoremcoe1subfv 16659 A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Poly1                            coe1 coe1coe1

Theoremcoe1mul2lem1 16660 An equivalence for coe1mul2 16662. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Theoremcoe1mul2lem2 16661* An equivalence for coe1mul2 16662. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Theoremcoe1mul2 16662* The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1                            coe1 g coe1 coe1

Theoremcoe1mul 16663* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1                            coe1 g coe1 coe1

Theoremply1tmcl 16664 Closure of the expression for a univariate polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g

Theoremcoe1tm 16665* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g       coe1

Theoremcoe1tmfv1 16666 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g       coe1

Theoremcoe1tmfv2 16667 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g                                          coe1

Theoremcoe1tmmul2 16668* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g                                                        coe1 coe1

Theoremcoe1tmmul 16669* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       var1              mulGrp       .g                                                        coe1 coe1

Theoremcoe1tmmul2fv 16670 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g                                                               coe1 coe1

Theoremcoe1pwmul 16671* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       var1       mulGrp       .g                                          coe1 coe1

Theoremcoe1pwmulfv 16672 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       var1       mulGrp       .g                                                 coe1 coe1

Theoremply1scltm 16673 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       var1              mulGrp       .g       algSc

Theoremcoe1sclmul 16674 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1                     algSc                     coe1 coe1

Theoremcoe1sclmulfv 16675 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1                     algSc                     coe1 coe1

Theoremcoe1sclmul2 16676 Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1                     algSc                     coe1 coe1

Theoremply1sclf 16677 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       algSc

Theoremcoe1scl 16678* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       algSc                     coe1

Theoremply1sclid 16679 Recover the base scalar from a scalar polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       algSc              coe1

Theoremply1sclf1 16680 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       algSc

Theoremply1scl0 16681 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       algSc

Theoremply1scln0 16682 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       algSc

Theoremply1scl1 16683 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       algSc

Theoremply1coe 16684* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Poly1       var1                     mulGrp       .g       coe1              g

10.11  The complex numbers as an extensible structure

10.11.1  Definition and basic properties

Syntaxcpsmet 16685 Extend class notation with the class of all pseudometric spaces.
PsMet

Syntaxcxmt 16686 Extend class notation with the class of all extended metric spaces.

Syntaxcme 16687 Extend class notation with the class of all metrics.

Syntaxcbl 16688 Extend class notation with the metric space ball function.

Syntaxcfbas 16689 Extend class definition to include the class of filter bases.

Syntaxcfg 16690 Extend class definition to include the filter generating function.

Syntaxcmopn 16691 Extend class notation with a function mapping each metric space to the family of its open sets.

SyntaxcmetuOLD 16692 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
metUnifOLD

Syntaxcmetu 16693 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
metUnif

Definitiondf-psmet 16694* Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Definitiondf-xmet 16695* Define the set of all extended metrics on a given base set. The definition is similar to df-met 16696, but we also allow the metric to take on the value . (Contributed by Mario Carneiro, 20-Aug-2015.)

Definitiondf-met 16696* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 18351. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. The 4 properties in Gleason's definition are shown by met0 18373, metgt0 18389, metsym 18380, and mettri 18382. (Contributed by NM, 25-Aug-2006.)

Definitiondf-bl 16697* Define the metric space ball function. See blval 18416 for its value. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Definitiondf-mopn 16698 Define a function whose value is the family of open sets of a metric space. See elmopn 18472 for its main property. (Contributed by NM, 1-Sep-2006.)

Definitiondf-fbas 16699* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)

Definitiondf-fg 16700* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)

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