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Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdeg1nn0clb 20001 A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1

Theoremdeg1lt0 20002 A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.)
deg1        Poly1

Theoremdeg1ldg 20003 A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                            coe1

Theoremdeg1ldgn 20004 An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                            coe1

Theoremdeg1ldgdomn 20005 A nonzero univariate polynomial over a domain always has a non-zero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Poly1                     RLReg       coe1       Domn

Theoremdeg1leb 20006* Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                     coe1

Theoremdeg1val 20007 Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.)
deg1        Poly1                     coe1

Theoremdeg1lt 20008 If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                     coe1

Theoremdeg1ge 20009 Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                     coe1

Theoremcoe1mul3 20010 The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1                            deg1                                                         coe1 coe1 coe1

Theoremcoe1mul4 20011 Value of the "leading" coefficient of a product of two nonzero polynomials. This will fail to actually be the leading coefficient only if it is zero (requiring the basic ring to contain zero divisors). (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1                            deg1                                                  coe1 coe1 coe1

Theoremdeg1addle 20012 The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1       deg1

Theoremdeg1addle2 20013 If both factors have degree bounded by , then the sum of the polynomials also has degree bounded by . (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       deg1

Theoremdeg1add 20014 Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       deg1

Theoremdeg1vscale 20015 The degree of a scalar times a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1       deg1

Theoremdeg1vsca 20016 The degree of a scalar times a polynomial is exactly the degree of the original polynomial when the scalar is not a zero divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       deg1                      RLReg

Theoremdeg1invg 20017 The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       deg1

Theoremdeg1suble 20018 The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1       deg1

Theoremdeg1sub 20019 Exact degree of a difference of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       deg1

Theoremdeg1mulle2 20020 Produce a bound on the product of two univariate polynomials given bounds on the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1       deg1

Theoremdeg1sublt 20021 Subtraction of two polynomials limited to the same degree with the same leading coefficient gives a polynomial with a smaller degree. (Contributed by Stefan O'Rear, 26-Mar-2015.)
deg1        Poly1                                                               coe1       coe1       coe1 coe1

Theoremdeg1le0 20022 A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
deg1        Poly1              algSc       coe1

Theoremdeg1sclle 20023 A scalar polynomial has nonpositive degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
deg1        Poly1              algSc

Theoremdeg1scl 20024 A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
deg1        Poly1              algSc

Theoremdeg1mul2 20025 Degree of multiplication of two nonzero polynomials when the first leads with a non-zero-divisor coefficient. (Contributed by Stefan O'Rear, 26-Mar-2015.)
deg1        Poly1       RLReg                                                 coe1

Theoremdeg1mul3 20026 Degree of multiplication of a polynomial on the left by a non-zero-dividing scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
deg1        Poly1       RLReg                     algSc

Theoremdeg1mul3le 20027 Degree of multiplication of a polynomial on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
deg1        Poly1                            algSc

Theoremdeg1tmle 20028 Limiting degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
deg1               Poly1       var1              mulGrp       .g

Theoremdeg1tm 20029 Exact degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
deg1               Poly1       var1              mulGrp       .g

Theoremdeg1pwle 20030 Limiting degree of a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
deg1        Poly1       var1       mulGrp       .g

Theoremdeg1pw 20031 Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.)
deg1        Poly1       var1       mulGrp       .g       NzRing

Theoremply1nz 20032 Univariate polynomials over a nonzero ring are a nonzero ring. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       NzRing NzRing

Theoremply1nzb 20033 Univariate polynomials are nonzero iff the base is nonzero. Or in contraposition, the univariate polynomials over the zero ring are also zero. (Contributed by Mario Carneiro, 13-Jun-2015.)
Poly1       NzRing NzRing

Theoremply1domn 20034 Corollary of deg1mul2 20025: the univariate polynomials over a domain are a domain. This is true for multivariate but with a much more complicated proof. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       Domn Domn

Theoremply1idom 20035 The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       IDomn IDomn

13.1.3  The division algorithm for univariate polynomials

Syntaxcmn1 20036 Monic polynomials.
Monic1p

Syntaxcuc1p 20037 Unitic polynomials.
Unic1p

Syntaxcq1p 20038 Univariate polynomial quotient.
quot1p

Syntaxcr1p 20039 Univariate polynomial remainder.
rem1p

Syntaxcig1p 20040 Univariate polynomial ideal generator.
idlGen1p

Definitiondf-mon1 20041* Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Monic1p Poly1 Poly1 coe1 deg1

Definitiondf-uc1p 20042* Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 20048. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Unic1p Poly1 Poly1 coe1 deg1 Unit

Definitiondf-q1p 20043* Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 20048. We actually use the reversed version for better harmony with our divisibility df-dvdsr 15734. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p Poly1 deg1 deg1

Definitiondf-r1p 20044* Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p Poly1 Poly1quot1pPoly1

Definitiondf-ig1p 20045* Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
idlGen1p LIdealPoly1 Poly1 Poly1 Monic1p deg1 deg1 Poly1

Theoremply1divmo 20046* Uniqueness of a quotient in a polynomial division. For polynomials such that and the leading coefficient of is not a zero divisor, there is at most one polynomial which satisfies where the degree of is less than the degree of . (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by NM, 17-Jun-2017.)
Poly1       deg1                                                                coe1        RLReg

Theoremply1divex 20047* Lemma for ply1divalg 20048: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       deg1                                                                                            coe1

Theoremply1divalg 20048* The division algorithm for univariate polynomials over a ring. For polynomials such that and the leading coefficient of is a unit, there are unique polynomials and such that the degree of is less than the degree of . (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       deg1                                                                coe1        Unit

Theoremply1divalg2 20049* Reverse the order of multiplication in ply1divalg 20048 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       deg1                                                                coe1        Unit

Theoremuc1pval 20050* Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1                     deg1        Unic1p       Unit       coe1

Theoremisuc1p 20051 Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1                     deg1        Unic1p       Unit       coe1

Theoremmon1pval 20052* Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1                     deg1        Monic1p              coe1

Theoremismon1p 20053 Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1                     deg1        Monic1p              coe1

Theoremuc1pcl 20054 Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1              Unic1p

Theoremmon1pcl 20055 Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1              Monic1p

Theoremuc1pn0 20056 Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1              Unic1p

Theoremmon1pn0 20057 Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1              Monic1p

Theoremuc1pdeg 20058 Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Unic1p

Theoremuc1pldg 20059 Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Unit       Unic1p       coe1

Theoremmon1pldg 20060 Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1               Monic1p       coe1

Theoremmon1puc1p 20061 Monic polynomials are unitic. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Unic1p       Monic1p

Theoremuc1pmon1p 20062 Make a unitic polynomial monic by multiplying a factor to normalize the leading coefficient. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Unic1p       Monic1p       Poly1              algSc       deg1               coe1

Theoremdeg1submon1p 20063 The difference of two monic polynomials of the same degree is a polynomial of lesser degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Monic1p       Poly1

Theoremq1pval 20064* Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p       Poly1              deg1

Theoremq1peqb 20065 Characterizing property of the polynomial quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p       Poly1              deg1                      Unic1p

Theoremq1pcl 20066 Closure of the quotient by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p       Poly1              Unic1p

Theoremr1pval 20067 Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p       Poly1              quot1p

Theoremr1pcl 20068 Closure of remainder following division by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p       Poly1              Unic1p

Theoremr1pdeglt 20069 The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p       Poly1              Unic1p       deg1

Theoremr1pid 20070 Express the original polynomial as using the quotient and remainder functions for and . (Contributed by Mario Carneiro, 5-Jun-2015.)
Poly1              Unic1p       quot1p       rem1p

Theoremdvdsq1p 20071 Divisibility in a polynomial ring is witnessed by the quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       r              Unic1p              quot1p

Theoremdvdsr1p 20072 Divisibility in a polynomial ring in terms of the remainder. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       r              Unic1p              rem1p

Theoremply1remlem 20073 A term of the form is linear, monic, and has exactly one zero. (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1                     var1              algSc              eval1       NzRing                     Monic1p       deg1

Theoremply1rem 20074 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 13030). If a polynomial is divided by the linear factor , the remainder is equal to , the evaluation of the polynomial at (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1                     var1              algSc              eval1       NzRing                            rem1p

Theoremfacth1 20075 The factor theorem and its converse. A polynomial has a root at iff is a factor of . (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1                     var1              algSc              eval1       NzRing                                   r

Theoremfta1glem1 20076 Lemma for fta1g 20078. (Contributed by Mario Carneiro, 7-Jun-2016.)
Poly1              deg1        eval1                     IDomn                     var1              algSc                                   quot1p

Theoremfta1glem2 20077* Lemma for fta1g 20078. (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1              deg1        eval1                     IDomn                     var1              algSc

Theoremfta1g 20078 The one-sided fundamental theorem of algebra. A polynomial of degree has at most roots. Unlike the real fundamental theorem fta 20850, which is only true in and other algebraically closed fields, this is true in any integral domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1              deg1        eval1                     IDomn

Theoremfta1blem 20079 Lemma for fta1b 20080. (Contributed by Mario Carneiro, 14-Jun-2015.)
Poly1              deg1        eval1                                   var1

Theoremfta1b 20080* The assumption that be a domain in fta1g 20078 is necessary. Here we show that the statement is strong enough to prove that is a domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1              deg1        eval1                     IDomn NzRing

Theoremdrnguc1p 20081 Over a division ring, all nonzero polynomials are unitic. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1                     Unic1p

Theoremig1peu 20082* There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       LIdeal              Monic1p       deg1

Theoremig1pval 20083* Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p              LIdeal       deg1        Monic1p

Theoremig1pval2 20084 Generator of the zero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p

Theoremig1pval3 20085 Characterizing properties of the monic generator of a nonzero ideal of polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p              LIdeal       deg1        Monic1p

Theoremig1pcl 20086 The monic generator of an ideal is always in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p       LIdeal

Theoremig1pdvds 20087 The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p       LIdeal       r

Theoremig1prsp 20088 Any ideal of polynomials over a division ring is generated by the ideal's canonical generator. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p       LIdeal       RSpan

Theoremply1lpir 20089 The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       LPIR

Theoremply1pid 20090 The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       Field PID

13.1.4  Elementary properties of complex polynomials

Syntaxcply 20091 Extend class notation to include the set of complex polynomials.
Poly

Syntaxcidp 20092 Extend class notation to include the identity polynomial.

Syntaxccoe 20093 Extend class notation to include the coefficient function on polynomials.
coeff

Syntaxcdgr 20094 Extend class notation to include the degree function on polynomials.
deg

Definitiondf-ply 20095* Define the set of polynomials on the complexes with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Definitiondf-idp 20096 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)

Definitiondf-coe 20097* Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff Poly

Definitiondf-dgr 20098 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg Poly coeff

Theoremplyco0 20099* Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplyval 20100* Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

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