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

Poly       Poly                                                                      Poly

Theoremplymullem 20002* Lemma for plymul 20004. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                                                                             Poly

Theoremplyadd 20003* The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly              Poly

Theoremplymul 20004* The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                     Poly

Theoremplysub 20005* The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                            Poly

Theoremplyaddcl 20006 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremplymulcl 20007 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremplysubcl 20008 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremcoeval 20009* Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly coeff

Theoremcoeeulem 20010* Lemma for coeeu 20011. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremcoeeu 20011* Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly

Theoremcoelem 20012* Lemma for properties of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly coeff coeff coeff

Theoremcoeeq 20013* If satisfies the properties of the coefficient function, it must be equal to the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly                                   coeff

Theoremdgrval 20014 Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly deg

Theoremdgrlem 20015* Lemma for dgrcl 20019 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef 20016 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef2 20017 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef3 20018 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremdgrcl 20019 The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly deg

Theoremdgrub 20020 If the -th coefficient of is nonzero, then the degree of is at least . (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       deg       Poly

Theoremdgrub2 20021 All the coefficients above the degree of are zero. (Contributed by Mario Carneiro, 23-Jul-2014.)
coeff       deg       Poly

Theoremdgrlb 20022 If all the coefficients above are zero, then the degree of is at most . (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       deg       Poly

Theoremcoeidlem 20023* Lemma for coeid 20024. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       deg       Poly

Theoremcoeid 20024* Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       deg       Poly

Theoremcoeid2 20025* Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       deg       Poly

Theoremcoeid3 20026* Reconstruct a polynomial as an explicit sum of the coefficient function up to at least the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       deg       Poly

Theoremplyco 20027* The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly       Poly                     Poly

Theoremcoeeq2 20028* Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly                            coeff

Theoremdgrle 20029* Given an explicit expression for a polynomial, the degree is at most the highest term in the sum. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly                            deg

Theoremdgreq 20030* If the highest term in a polynomial expression is nonzero, then the polynomial's degree is completely determined. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly                                          deg

Theorem0dgr 20031 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg

Theorem0dgrb 20032 A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly deg

Theoremcoefv0 20033 The result of evaluating a polynomial at zero is the constant term. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       Poly

coeff       coeff       deg       deg       Poly Poly coeff deg

Theoremcoemullem 20035* Lemma for coemul 20037 and dgrmul 20055. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       deg       deg       Poly Poly coeff deg

Theoremcoeadd 20036 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoemul 20037* A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoe11 20038 The coefficient function is one-to-one, so if the coefficients are equal then the functions are equal and vice-versa. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
coeff       coeff       Poly Poly

Theoremcoemulhi 20039 The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       deg       deg       Poly Poly coeff

Theoremcoemulc 20040 The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly coeff coeff

Theoremcoe0 20041 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff

Theoremcoesub 20042 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoe1termlem 20043* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
coeff deg

Theoremcoe1term 20044* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
coeff

Theoremdgr1term 20045* The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
deg

Theoremplycn 20046 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly

Theoremdgr0 20047 The degree of the zero polynomial is zero. Note: this differs from some other definitions of the degree of the zero polynomial, such as or undefined. But it is convenient for us to define it this way, so that we have dgrcl 20019, dgreq0 20050 and coeid 20024 without having to special-case zero, although plydivalg 20083 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg

Theoremcoeidp 20048 The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
coeff

Theoremdgrid 20049 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
deg

Theoremdgreq0 20050 The leading coefficient of a polynomial is nonzero, unless the entire polynomial is zero. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Fan Zheng, 21-Jun-2016.)
deg       coeff       Poly

Theoremdgrlt 20051 Two ways to say that the degree of is strictly less than . (Contributed by Mario Carneiro, 25-Jul-2014.)
deg       coeff       Poly

Theoremdgradd 20052 The degree of a sum of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg       deg       Poly Poly deg

Theoremdgradd2 20053 The degree of a sum of polynomials of unequal degrees is the degree of the larger polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg       deg       Poly Poly deg

Theoremdgrmul2 20054 The degree of a product of polynomials is at most the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg       deg       Poly Poly deg

Theoremdgrmul 20055 The degree of a product of nonzero polynomials is the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg       deg       Poly Poly deg

Theoremdgrmulc 20056 Scalar multiplication by a nonzero constant does not change the degree of a function. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly deg deg

Theoremdgrsub 20057 The degree of a difference of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 26-Jul-2014.)
deg       deg       Poly Poly deg

Theoremdgrcolem1 20058* The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)
deg                     Poly       deg

Theoremdgrcolem2 20059* Lemma for dgrco 20060. (Contributed by Mario Carneiro, 15-Sep-2014.)
deg       deg       Poly       Poly       coeff                     Polydeg deg deg        deg

Theoremdgrco 20060 The degree of a composition of two polynomials is the product of the degrees. (Contributed by Mario Carneiro, 15-Sep-2014.)
deg       deg       Poly       Poly       deg

Theoremplycjlem 20061* Lemma for plycj 20062 and coecj 20063. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg              coeff       Poly

Theoremplycj 20062* The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on independently of .) (Contributed by Mario Carneiro, 24-Jul-2014.)
deg                     Poly       Poly

Theoremcoecj 20063 Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg              coeff       Poly coeff

Theoremplyrecj 20064 A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly

Theoremplymul0or 20065 Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly Poly

Theoremofmulrt 20066 The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremplyreres 20067 Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Poly

Theoremdvply1 20068* Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvply2g 20069 The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.)
SubRingfld Poly Poly

Theoremdvply2 20070 The derivative of a polynomial is a polynomial. (Contributed by Stefan O'Rear, 14-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
Poly Poly

Theoremdvnply2 20071 Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.)
SubRingfld Poly Poly

Theoremdvnply 20072 Polynomials have polynomials as derivatives of all orders. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
Poly Poly

Theoremplycpn 20073 Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
Poly

13.1.5  The division algorithm for polynomials

Syntaxcquot 20074 Extend class notation to include the quotient of a polynomial division.
quot

Definitiondf-quot 20075* Define the quotient function on polynomials. This is the of the expression in the division algorithm. (Contributed by Mario Carneiro, 23-Jul-2014.)
quot Poly Poly Poly deg deg

Theoremquotval 20076* Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly Poly quot Poly deg deg

Theoremplydivlem1 20077* Lemma for plydivalg 20083. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremplydivlem2 20078* Lemma for plydivalg 20083. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly       Poly                     Poly Poly

Theoremplydivlem3 20079* Lemma for plydivex 20081. Base case of induction. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly       Poly                     deg deg        Poly deg deg

Theoremplydivlem4 20080* Lemma for plydivex 20081. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly       Poly                                                        Poly deg Poly deg        coeff       coeff       deg       deg       Poly deg

Theoremplydivex 20081* Lemma for plydivalg 20083. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly       Poly                     Poly deg deg

Theoremplydiveu 20082* Lemma for plydivalg 20083. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly       Poly                     Poly       deg deg              Poly       deg deg

Theoremplydivalg 20083* The division algorithm on polynomials over a subfield of the complex numbers. If and are polynomials over , then there is a unique quotient polynomial such that the remainder is either zero or has degree less than . (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly       Poly                     Poly deg deg

Theoremquotlem 20084* Lemma for properties of the polynomial quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly       Poly              quot        quot Poly deg deg

Theoremquotcl 20085* The quotient of two polynomials in a field is also in the field. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly       Poly              quot Poly

Theoremquotcl2 20086 Closure of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly Poly quot Poly

Theoremquotdgr 20087 Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
quot        Poly Poly deg deg

Theoremplyremlem 20088 Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly deg

Theoremplyrem 20089 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12969). 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, 26-Jul-2014.)
quot        Poly

Theoremfacth 20090 The factor theorem. If a polynomial has a root at , then is a factor of (and the other factor is quot ). (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly quot

Theoremfta1lem 20091* Lemma for fta1 20092. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly        deg               Poly deg deg       deg

Theoremfta1 20092 The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg roots. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly deg

Theoremquotcan 20093 Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly Poly quot

Theoremvieta1lem1 20094* Lemma for vieta1 20096. (Contributed by Mario Carneiro, 28-Jul-2014.)
coeff       deg              Poly                            Poly deg deg coeffdeg coeffdeg       quot        Poly deg

Theoremvieta1lem2 20095* Lemma for vieta1 20096: inductive step. Let be a root of . Then for some by the factor theorem, and is a degree- polynomial, so by the induction hypothesis coeff coeff, so coeff coeff. Now the coefficients of are coeff and coeff coeff , which works out to coeff coeff , so putting it all together we have as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.)
coeff       deg              Poly                            Poly deg deg coeffdeg coeffdeg       quot

Theoremvieta1 20096* The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree has distinct roots, then the sum over these roots can be calculated as . (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (Contributed by Mario Carneiro, 28-Jul-2014.)
coeff       deg              Poly

Theoremplyexmo 20097* An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)
Poly

13.1.6  Algebraic numbers

Syntaxcaa 20098 Extend class notation to include the set of algebraic numbers.

Definitiondf-aa 20099 Define the set of algebraic numbers. An algebraic number is a root of a nonzero polynomial over the integers. Here we construct it as the union of all kernels (preimages of ) of all polynomials in Poly, except the zero polynomial . (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremelaa 20100* Elementhood in the set of algebraic numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

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