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

Theoremig1pdvds 20101 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 20102 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 20103 The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       LPIR

Theoremply1pid 20104 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 20105 Extend class notation to include the set of complex polynomials.
Poly

Syntaxcidp 20106 Extend class notation to include the identity polynomial.

Syntaxccoe 20107 Extend class notation to include the coefficient function on polynomials.
coeff

Syntaxcdgr 20108 Extend class notation to include the degree function on polynomials.
deg

Definitiondf-ply 20109* Define the set of polynomials on the complexes with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Definitiondf-idp 20110 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)

Definitiondf-coe 20111* Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff Poly

Definitiondf-dgr 20112 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg Poly coeff

Theoremplyco0 20113* Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplyval 20114* Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplybss 20115 Reverse closure of the parameter of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremelply 20116* Definition of a polynomial with coefficients in . (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremelply2 20117* 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 20118 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 20119 The polynomial is a function on the complexes. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremplyss 20120 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly Poly

Theoremplyssc 20121 Every polynomial ring is contained in the ring of polynomials over . (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly Poly

Theoremelplyr 20122* 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 20123* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremply1termlem 20124* Lemma for ply1term 20125. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremply1term 20125* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplypow 20126* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyconst 20127 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremne0p 20128 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theoremply0 20129 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyid 20130 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyeq0lem 20131* Lemma for plyeq0 20132. 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 20132* 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 20111 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplypf1 20133 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 20134* Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly       Poly

Theoremplymullem1 20135* Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly       Poly

Theoremplyaddlem 20136* Lemma for plyadd 20138. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                                                                      Poly

Theoremplymullem 20137* Lemma for plymul 20139. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                                                                             Poly

Theoremplyadd 20138* The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly              Poly

Theoremplymul 20139* The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                     Poly

Theoremplysub 20140* The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                            Poly

Theoremplyaddcl 20141 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremplymulcl 20142 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremplysubcl 20143 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremcoeval 20144* Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly coeff

Theoremcoeeulem 20145* Lemma for coeeu 20146. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremcoeeu 20146* Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly

Theoremcoelem 20147* 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 20148* 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 20149 Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly deg

Theoremdgrlem 20150* Lemma for dgrcl 20154 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef 20151 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef2 20152 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremcoef3 20153 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly

Theoremdgrcl 20154 The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly deg

Theoremdgrub 20155 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 20156 All the coefficients above the degree of are zero. (Contributed by Mario Carneiro, 23-Jul-2014.)
coeff       deg       Poly

Theoremdgrlb 20157 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 20158* Lemma for coeid 20159. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       deg       Poly

Theoremcoeid 20159* 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 20160* 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 20161* 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 20162* 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 20163* Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly                            coeff

Theoremdgrle 20164* 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 20165* 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 20166 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg

Theorem0dgrb 20167 A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly deg

Theoremcoefv0 20168 The result of evaluating a polynomial at zero is the constant term. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       Poly

Theoremcoeaddlem 20169 Lemma for coeadd 20171 and dgradd 20187. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       deg       deg       Poly Poly coeff deg

Theoremcoemullem 20170* Lemma for coemul 20172 and dgrmul 20190. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       deg       deg       Poly Poly coeff deg

Theoremcoeadd 20171 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoemul 20172* A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoe11 20173 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 20174 The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       deg       deg       Poly Poly coeff

Theoremcoemulc 20175 The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly coeff coeff

Theoremcoe0 20176 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff

Theoremcoesub 20177 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
coeff       coeff       Poly Poly coeff

Theoremcoe1termlem 20178* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
coeff deg

Theoremcoe1term 20179* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
coeff

Theoremdgr1term 20180* The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
deg

Theoremplycn 20181 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly

Theoremdgr0 20182 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 20154, dgreq0 20185 and coeid 20159 without having to special-case zero, although plydivalg 20218 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg

Theoremcoeidp 20183 The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
coeff

Theoremdgrid 20184 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
deg

Theoremdgreq0 20185 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 20186 Two ways to say that the degree of is strictly less than . (Contributed by Mario Carneiro, 25-Jul-2014.)
deg       coeff       Poly

Theoremdgradd 20187 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 20188 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 20189 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 20190 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 20191 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 20192 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 20193* The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)
deg                     Poly       deg

Theoremdgrcolem2 20194* Lemma for dgrco 20195. (Contributed by Mario Carneiro, 15-Sep-2014.)
deg       deg       Poly       Poly       coeff                     Polydeg deg deg        deg

Theoremdgrco 20195 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 20196* Lemma for plycj 20197 and coecj 20198. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg              coeff       Poly

Theoremplycj 20197* 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 20198 Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.)
deg              coeff       Poly coeff

Theoremplyrecj 20199 A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly

Theoremplymul0or 20200 Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly Poly

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