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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremig1pval2 20101 Generator of the zero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p

Theoremig1pval3 20102 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 20103 The monic generator of an ideal is always in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p       LIdeal

Theoremig1pdvds 20104 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 20105 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 20106 The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       LPIR

Theoremply1pid 20107 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 20108 Extend class notation to include the set of complex polynomials.
Poly

Syntaxcidp 20109 Extend class notation to include the identity polynomial.

Syntaxccoe 20110 Extend class notation to include the coefficient function on polynomials.
coeff

Syntaxcdgr 20111 Extend class notation to include the degree function on polynomials.
deg

Definitiondf-ply 20112* Define the set of polynomials on the complexes with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Definitiondf-idp 20113 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)

Definitiondf-coe 20114* Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff Poly

Definitiondf-dgr 20115 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg Poly coeff

Theoremplyco0 20116* Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplyval 20117* Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplybss 20118 Reverse closure of the parameter of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremelply 20119* Definition of a polynomial with coefficients in . (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremelply2 20120* The coefficient function can be assumed to have zeroes outside . (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly

Theoremplyun0 20121 The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly Poly

Theoremplyf 20122 The polynomial is a function on the complexes. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremplyss 20123 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly Poly

Theoremplyssc 20124 Every polynomial ring is contained in the ring of polynomials over . (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly Poly

Theoremelplyr 20125* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly

Theoremelplyd 20126* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremply1termlem 20127* Lemma for ply1term 20128. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremply1term 20128* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplypow 20129* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyconst 20130 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremne0p 20131 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theoremply0 20132 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyid 20133 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyeq0lem 20134* Lemma for plyeq0 20135. If is the coefficient function for a nonzero polynomial such that for every and is the nonzero leading coefficient, then the function is a sum of powers of , and so the limit of this function as is the constant term, . But everywhere, so this limit is also equal to zero so that , a contradiction. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplyeq0 20135* If a polynomial is zero at every point (or even just zero at the positive integers), then all the coefficients must be zero. This is the basis for the method of equating coefficients of equal polynomials, and ensures that df-coe 20114 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplypf1 20136 Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.)
flds        Poly1              eval1fld       SubRingfld Poly

Theoremplyaddlem1 20137* Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly       Poly

Theoremplymullem1 20138* Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly       Poly

Poly       Poly                                                                      Poly

Theoremplymullem 20140* Lemma for plymul 20142. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                                                                             Poly

Theoremplyadd 20141* The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly              Poly

Theoremplymul 20142* The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                     Poly

Theoremplysub 20143* The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                            Poly

Theoremplyaddcl 20144 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremplymulcl 20145 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremplysubcl 20146 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremcoeval 20147* Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly coeff

Theoremcoeeulem 20148* Lemma for coeeu 20149. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremcoeeu 20149* Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly

Theoremcoelem 20150* 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 20151* 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 20152 Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly deg

Theoremdgrlem 20153* Lemma for dgrcl 20157 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef 20154 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef2 20155 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef3 20156 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremdgrcl 20157 The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly deg

Theoremdgrub 20158 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 20159 All the coefficients above the degree of are zero. (Contributed by Mario Carneiro, 23-Jul-2014.)
coeff       deg       Poly

Theoremdgrlb 20160 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 20161* Lemma for coeid 20162. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       deg       Poly

Theoremcoeid 20162* 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 20163* 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 20164* 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 20165* 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 20166* Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly                            coeff

Theoremdgrle 20167* 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 20168* 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 20169 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg

Theorem0dgrb 20170 A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly deg

Theoremcoefv0 20171 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 20173* Lemma for coemul 20175 and dgrmul 20193. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       deg       deg       Poly Poly coeff deg

Theoremcoeadd 20174 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoemul 20175* A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoe11 20176 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 20177 The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       deg       deg       Poly Poly coeff

Theoremcoemulc 20178 The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly coeff coeff

Theoremcoe0 20179 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff

Theoremcoesub 20180 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoe1termlem 20181* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
coeff deg

Theoremcoe1term 20182* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
coeff

Theoremdgr1term 20183* The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
deg

Theoremplycn 20184 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly

Theoremdgr0 20185 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 20157, dgreq0 20188 and coeid 20162 without having to special-case zero, although plydivalg 20221 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg

Theoremcoeidp 20186 The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
coeff

Theoremdgrid 20187 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
deg

Theoremdgreq0 20188 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 20189 Two ways to say that the degree of is strictly less than . (Contributed by Mario Carneiro, 25-Jul-2014.)
deg       coeff       Poly

Theoremdgradd 20190 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 20191 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 20192 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 20193 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 20194 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 20195 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 20196* The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)
deg                     Poly       deg

Theoremdgrcolem2 20197* Lemma for dgrco 20198. (Contributed by Mario Carneiro, 15-Sep-2014.)
deg       deg       Poly       Poly       coeff                     Polydeg deg deg        deg

Theoremdgrco 20198 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 20199* Lemma for plycj 20200 and coecj 20201. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg              coeff       Poly

Theoremplycj 20200* 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

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