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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcdgr 19401 Extend class notation to include the degree function on polynomials.
 class deg
 
Definitiondf-ply 19402* Define the set of polynomials on the complexes with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |- Poly  =  ( x  e.  ~P CC  |->  { f  |  E. n  e.  NN0  E. a  e.  ( ( x  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  ( z ^
 k ) ) ) } )
 
Definitiondf-idp 19403 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  X p  =  (  _I  |`  CC )
 
Definitiondf-coe 19404* Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |- coeff  =  ( f  e.  (Poly `  CC )  |->  ( iota_ a  e.  ( CC  ^m  NN0 ) E. n  e. 
 NN0  ( ( a
 " ( ZZ>= `  ( n  +  1 )
 ) )  =  {
 0 }  /\  f  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... n ) ( ( a `  k
 )  x.  ( z ^ k ) ) ) ) ) )
 
Definitiondf-dgr 19405 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |- deg 
 =  ( f  e.  (Poly `  CC )  |-> 
 sup ( ( `' (coeff `  f ) " ( CC  \  {
 0 } ) ) ,  NN0 ,  <  ) )
 
Theoremplyco0 19406* Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ( N  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( A `  k )  =/=  0  ->  k  <_  N )
 ) )
 
Theoremplyval 19407* Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( S  C_  CC  ->  (Poly `  S )  =  { f  |  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  ( z ^
 k ) ) ) } )
 
Theoremplybss 19408 Reverse closure of the parameter  S of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  S  C_ 
 CC )
 
Theoremelply 19409* Definition of a polynomial with coefficients in  S. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  <->  ( S  C_  CC  /\  E. n  e. 
 NN0  E. a  e.  (
 ( S  u.  {
 0 } )  ^m  NN0 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  ( z ^
 k ) ) ) ) )
 
Theoremelply2 19410* The coefficient function can be assumed to have zeroes outside  0 ... n. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( F  e.  (Poly `  S )  <->  ( S  C_  CC  /\  E. n  e. 
 NN0  E. a  e.  (
 ( S  u.  {
 0 } )  ^m  NN0 ) ( ( a
 " ( ZZ>= `  ( n  +  1 )
 ) )  =  {
 0 }  /\  F  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... n ) ( ( a `  k
 )  x.  ( z ^ k ) ) ) ) ) )
 
Theoremplyun0 19411 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 `  ( S  u.  { 0 } )
 )  =  (Poly `  S )
 
Theoremplyf 19412 The polynomial is a function on the complexes. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F : CC --> CC )
 
Theoremplyss 19413 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  T  /\  T  C_  CC )  ->  (Poly `  S )  C_  (Poly `  T ) )
 
Theoremplyssc 19414 Every polynomial ring is contained in the ring of polynomials over  CC. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (Poly `  S )  C_  (Poly `  CC )
 
Theoremelplyr 19415* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0 --> S ) 
 ->  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
 
Theoremelplyd 19416* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  ( 0
 ... N ) ) 
 ->  A  e.  S )   =>    |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  (
 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
 
Theoremply1termlem 19417* Lemma for ply1term 19418. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  F  =  ( z  e.  CC  |->  ( A  x.  ( z ^ N ) ) )   =>    |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ...
 N ) ( if ( k  =  N ,  A ,  0 )  x.  ( z ^
 k ) ) ) )
 
Theoremply1term 19418* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  F  =  ( z  e.  CC  |->  ( A  x.  ( z ^ N ) ) )   =>    |-  ( ( S  C_  CC  /\  A  e.  S  /\  N  e.  NN0 )  ->  F  e.  (Poly `  S ) )
 
Theoremplypow 19419* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  CC  /\  1  e.  S  /\  N  e.  NN0 )  ->  ( z  e.  CC  |->  ( z ^ N ) )  e.  (Poly `  S ) )
 
Theoremplyconst 19420 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  CC  /\  A  e.  S )  ->  ( CC  X.  { A } )  e.  (Poly `  S )
 )
 
Theoremne0p 19421 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ( A  e.  CC  /\  ( F `  A )  =/=  0
 )  ->  F  =/=  0 p )
 
Theoremply0 19422 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( S  C_  CC  ->  0 p  e.  (Poly `  S ) )
 
Theoremplyid 19423 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  CC  /\  1  e.  S )  ->  X p  e.  (Poly `  S ) )
 
Theoremplyeq0lem 19424* Lemma for plyeq0 19425. If  A is the coefficient function for a nonzero polynomial such that  P ( z )  =  sum_ k  e.  NN0 A ( k )  x.  z ^
k  =  0 for every  z  e.  CC and  A ( M ) is the nonzero leading coefficient, then the function  F ( z )  =  P ( z )  /  z ^ M is a sum of powers of  1  /  z, and so the limit of this function as  z 
~~>  +oo is the constant term,  A ( M ). But  F ( z )  =  0 everywhere, so this limit is also equal to zero so that  A ( M )  =  0, a contradiction. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( N  +  1 )
 ) )  =  {
 0 } )   &    |-  ( ph  ->  0 p  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  M  =  sup ( ( `' A " ( S  \  {
 0 } ) ) ,  RR ,  <  )   &    |-  ( ph  ->  ( `' A " ( S  \  { 0 } )
 )  =/=  (/) )   =>    |-  -.  ph
 
Theoremplyeq0 19425* 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 19404 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( N  +  1 )
 ) )  =  {
 0 } )   &    |-  ( ph  ->  0 p  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  A  =  ( NN0  X.  {
 0 } ) )
 
Theoremplypf1 19426 Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  R  =  (flds  S )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  (
 Base `  P )   &    |-  E  =  (eval1 ` fld )   =>    |-  ( S  e.  (SubRing ` fld ) 
 ->  (Poly `  S )  =  ( E " A ) )
 
Theoremplyaddlem1 19427* Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  B : NN0 --> CC )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  ( F  o F  +  G )  =  ( z  e.  CC  |->  sum_ k  e.  (
 0 ... if ( M 
 <_  N ,  N ,  M ) ) ( ( ( A  o F  +  B ) `  k )  x.  (
 z ^ k ) ) ) )
 
Theoremplymullem1 19428* Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  B : NN0 --> CC )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  ( F  o F  x.  G )  =  ( z  e.  CC  |->  sum_ n  e.  (
 0 ... ( M  +  N ) ) (
 sum_ k  e.  (
 0 ... n ) ( ( A `  k
 )  x.  ( B `
  ( n  -  k ) ) )  x.  ( z ^ n ) ) ) )
 
Theoremplyaddlem 19429* Lemma for plyadd 19431. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  B  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  ( F  o F  +  G )  e.  (Poly `  S ) )
 
Theoremplymullem 19430* Lemma for plymul 19432. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  B  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  x.  y )  e.  S )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e.  (Poly `  S ) )
 
Theoremplyadd 19431* The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   =>    |-  ( ph  ->  ( F  o F  +  G )  e.  (Poly `  S ) )
 
Theoremplymul 19432* The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  x.  y )  e.  S )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e.  (Poly `  S ) )
 
Theoremplysub 19433* The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  x.  y )  e.  S )   &    |-  ( ph  ->  -u 1  e.  S )   =>    |-  ( ph  ->  ( F  o F  -  G )  e.  (Poly `  S ) )
 
Theoremplyaddcl 19434 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  o F  +  G )  e.  (Poly `  CC ) )
 
Theoremplymulcl 19435 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  o F  x.  G )  e.  (Poly `  CC ) )
 
Theoremplysubcl 19436 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  o F  -  G )  e.  (Poly `  CC ) )
 
Theoremcoeval 19437* Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  (coeff `  F )  =  (
 iota_ a  e.  ( CC  ^m  NN0 ) E. n  e.  NN0  ( ( a
 " ( ZZ>= `  ( n  +  1 )
 ) )  =  {
 0 }  /\  F  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... n ) ( ( a `  k
 )  x.  ( z ^ k ) ) ) ) ) )
 
Theoremcoeeulem 19438* Lemma for coeeu 19439. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  A  e.  ( CC  ^m  NN0 ) )   &    |-  ( ph  ->  B  e.  ( CC  ^m  NN0 ) )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( A " ( ZZ>= `  ( M  +  1
 ) ) )  =  { 0 } )   &    |-  ( ph  ->  ( B "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremcoeeu 19439* Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( F  e.  (Poly `  S )  ->  E! a  e.  ( CC  ^m 
 NN0 ) E. n  e.  NN0  ( ( a
 " ( ZZ>= `  ( n  +  1 )
 ) )  =  {
 0 }  /\  F  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... n ) ( ( a `  k
 )  x.  ( z ^ k ) ) ) ) )
 
Theoremcoelem 19440* Lemma for properties of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( F  e.  (Poly `  S )  ->  (
 (coeff `  F )  e.  ( CC  ^m  NN0 )  /\  E. n  e. 
 NN0  ( ( (coeff `  F ) " ( ZZ>=
 `  ( n  +  1 ) ) )  =  { 0 } 
 /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F ) `  k )  x.  (
 z ^ k ) ) ) ) ) )
 
Theoremcoeeq 19441* If  A 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.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  ( A "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  (coeff `  F )  =  A )
 
Theoremdgrval 19442 Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  (deg `  F )  =  sup ( ( `' A " ( CC  \  {
 0 } ) ) ,  NN0 ,  <  ) )
 
Theoremdgrlem 19443* Lemma for dgrcl 19447 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A "
 ( CC  \  {
 0 } ) ) x  <_  n )
 )
 
Theoremcoef 19444 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  A : NN0 --> ( S  u.  { 0 } ) )
 
Theoremcoef2 19445 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  A  =  (coeff `  F )   =>    |-  ( ( F  e.  (Poly `  S )  /\  0  e.  S )  ->  A : NN0 --> S )
 
Theoremcoef3 19446 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  A : NN0 --> CC )
 
Theoremdgrcl 19447 The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  (deg `  F )  e.  NN0 )
 
Theoremdgrub 19448 If the  M-th coefficient of  F is nonzero, then the degree of  F is at least  M. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   =>    |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `  M )  =/=  0 )  ->  M  <_  N )
 
Theoremdgrub2 19449 All the coefficients above the degree of  F are zero. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  ( A " ( ZZ>= `  ( N  +  1 )
 ) )  =  {
 0 } )
 
Theoremdgrlb 19450 If all the coefficients above  M are zero, then the degree of  F is at most  M. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   =>    |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A " ( ZZ>= `  ( M  +  1 )
 ) )  =  {
 0 } )  ->  N  <_  M )
 
Theoremcoeidlem 19451* Lemma for coeid 19452. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  B  e.  ( ( S  u.  { 0 } )  ^m  NN0 )
 )   &    |-  ( ph  ->  ( B " ( ZZ>= `  ( M  +  1 )
 ) )  =  {
 0 } )   &    |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ...
 M ) ( ( B `  k )  x.  ( z ^
 k ) ) ) )   =>    |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )
 
Theoremcoeid 19452* 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.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  F  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )
 
Theoremcoeid2 19453* 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.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   =>    |-  ( ( F  e.  (Poly `  S )  /\  X  e.  CC )  ->  ( F `  X )  =  sum_ k  e.  ( 0 ... N ) ( ( A `
  k )  x.  ( X ^ k
 ) ) )
 
Theoremcoeid3 19454* 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.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   =>    |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( F `
  X )  = 
 sum_ k  e.  (
 0 ... M ) ( ( A `  k
 )  x.  ( X ^ k ) ) )
 
Theoremplyco 19455* The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  x.  y )  e.  S )   =>    |-  ( ph  ->  ( F  o.  G )  e.  (Poly `  S )
 )
 
Theoremcoeeq2 19456* Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  ( 0
 ... N ) ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ...
 N ) ( A  x.  ( z ^
 k ) ) ) )   =>    |-  ( ph  ->  (coeff `  F )  =  ( k  e.  NN0  |->  if (
 k  <_  N ,  A ,  0 )
 ) )
 
Theoremdgrle 19457* 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.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  ( 0
 ... N ) ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ...
 N ) ( A  x.  ( z ^
 k ) ) ) )   =>    |-  ( ph  ->  (deg `  F )  <_  N )
 
Theoremdgreq 19458* 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.)
 |-  ( ph  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  ( A "
 ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )   &    |-  ( ph  ->  F  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  ( A `  N )  =/=  0 )   =>    |-  ( ph  ->  (deg `  F )  =  N )
 
Theorem0dgr 19459 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( A  e.  CC  ->  (deg `  ( CC  X. 
 { A } )
 )  =  0 )
 
Theorem0dgrb 19460 A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  (
 (deg `  F )  =  0  <->  F  =  ( CC  X.  { ( F `
  0 ) }
 ) ) )
 
Theoremcoefv0 19461 The result of evaluating a polynomial at zero is the constant term. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  ( F `  0 )  =  ( A `  0
 ) )
 
Theoremcoeaddlem 19462 Lemma for coeadd 19464 and dgradd 19480. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   &    |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( (coeff `  ( F  o F  +  G ) )  =  ( A  o F  +  B )  /\  (deg `  ( F  o F  +  G ) )  <_  if ( M  <_  N ,  N ,  M ) ) )
 
Theoremcoemullem 19463* Lemma for coemul 19465 and dgrmul 19483. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   &    |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( (coeff `  ( F  o F  x.  G ) )  =  ( n  e.  NN0  |->  sum_
 k  e.  ( 0
 ... n ) ( ( A `  k
 )  x.  ( B `
  ( n  -  k ) ) ) )  /\  (deg `  ( F  o F  x.  G ) )  <_  ( M  +  N ) ) )
 
Theoremcoeadd 19464 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  (coeff `  ( F  o F  +  G ) )  =  ( A  o F  +  B ) )
 
Theoremcoemul 19465* A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  ( (coeff `  ( F  o F  x.  G ) ) `  N )  =  sum_ k  e.  ( 0 ...
 N ) ( ( A `  k )  x.  ( B `  ( N  -  k
 ) ) ) )
 
Theoremcoe11 19466 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.)
 |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  =  G  <->  A  =  B ) )
 
Theoremcoemulhi 19467 The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   &    |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( (coeff `  ( F  o F  x.  G ) ) `  ( M  +  N ) )  =  (
 ( A `  M )  x.  ( B `  N ) ) )
 
Theoremcoemulc 19468 The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  ->  (coeff `  ( ( CC 
 X.  { A } )  o F  x.  F ) )  =  (
 ( NN0  X.  { A } )  o F  x.  (coeff `  F )
 ) )
 
Theoremcoe0 19469 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (coeff `  0 p
 )  =  ( NN0  X. 
 { 0 } )
 
Theoremcoesub 19470 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  (coeff `  ( F  o F  -  G ) )  =  ( A  o F  -  B ) )
 
Theoremcoe1termlem 19471* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  F  =  ( z  e.  CC  |->  ( A  x.  ( z ^ N ) ) )   =>    |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( (coeff `  F )  =  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 )
 )  /\  ( A  =/=  0  ->  (deg `  F )  =  N ) ) )
 
Theoremcoe1term 19472* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  F  =  ( z  e.  CC  |->  ( A  x.  ( z ^ N ) ) )   =>    |-  ( ( A  e.  CC  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( (coeff `  F ) `  M )  =  if ( M  =  N ,  A , 
 0 ) )
 
Theoremdgr1term 19473* The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  F  =  ( z  e.  CC  |->  ( A  x.  ( z ^ N ) ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  NN0 )  ->  (deg `  F )  =  N )
 
Theoremplycn 19474 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F  e.  ( CC -cn-> CC )
 )
 
Theoremdgr0 19475 The degree of the zero polynomial is zero. Note: this differs from some other definitions of the degree of the zero polynomial, such as  -u 1 , 
-oo or undefined. But it is convenient for us to define it this way, so that we have dgrcl 19447, dgreq0 19478 and coeid 19452 without having to special-case zero, although plydivalg 19511 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (deg `  0 p
 )  =  0
 
Theoremcoeidp 19476 The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( A  e.  NN0  ->  ( (coeff `  X p
 ) `  A )  =  if ( A  =  1 ,  1 , 
 0 ) )
 
Theoremdgrid 19477 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  (deg `  X p
 )  =  1
 
Theoremdgreq0 19478 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.)
 |-  N  =  (deg `  F )   &    |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  ( F  =  0 p  <->  ( A `  N )  =  0 ) )
 
Theoremdgrlt 19479 Two ways to say that the degree of 
F is strictly less than 
N. (Contributed by Mario Carneiro, 25-Jul-2014.)
 |-  N  =  (deg `  F )   &    |-  A  =  (coeff `  F )   =>    |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( ( F  =  0 p  \/  N  <  M )  <->  ( N  <_  M 
 /\  ( A `  M )  =  0
 ) ) )
 
Theoremdgradd 19480 The degree of a sum of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  (deg `  ( F  o F  +  G ) )  <_  if ( M  <_  N ,  N ,  M ) )
 
Theoremdgradd2 19481 The degree of a sum of polynomials of unequal degrees is the degree of the larger polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G ) )  =  N )
 
Theoremdgrmul2 19482 The degree of a product of polynomials is at most the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  (deg `  ( F  o F  x.  G ) )  <_  ( M  +  N ) )
 
Theoremdgrmul 19483 The degree of a product of nonzero polynomials is the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( F  o F  x.  G ) )  =  ( M  +  N ) )
 
Theoremdgrmulc 19484 Scalar multiplication by a nonzero constant does not change the degree of a function. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  ->  (deg `  ( ( CC 
 X.  { A } )  o F  x.  F ) )  =  (deg `  F ) )
 
Theoremdgrsub 19485 The degree of a difference of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  (deg `  ( F  o F  -  G ) )  <_  if ( M  <_  N ,  N ,  M ) )
 
Theoremdgrcolem1 19486* The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  N  =  (deg `  G )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   =>    |-  ( ph  ->  (deg `  ( x  e.  CC  |->  ( ( G `  x ) ^ M ) ) )  =  ( M  x.  N ) )
 
Theoremdgrcolem2 19487* Lemma for dgrco 19488. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  G  e.  (Poly `  S ) )   &    |-  A  =  (coeff `  F )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  M  =  ( D  +  1 ) )   &    |-  ( ph  ->  A. f  e.  (Poly `  CC ) ( (deg `  f )  <_  D  ->  (deg `  ( f  o.  G ) )  =  ( (deg `  f
 )  x.  N ) ) )   =>    |-  ( ph  ->  (deg `  ( F  o.  G ) )  =  ( M  x.  N ) )
 
Theoremdgrco 19488 The degree of a composition of two polynomials is the product of the degrees. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  G  e.  (Poly `  S ) )   =>    |-  ( ph  ->  (deg `  ( F  o.  G ) )  =  ( M  x.  N ) )
 
Theoremplycjlem 19489* Lemma for plycj 19490 and coecj 19491. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  N  =  (deg `  F )   &    |-  G  =  ( ( *  o.  F )  o.  * )   &    |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ...
 N ) ( ( ( *  o.  A ) `  k )  x.  ( z ^ k
 ) ) ) )
 
Theoremplycj 19490* 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  ( * `  z ) independently of  z.) (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  N  =  (deg `  F )   &    |-  G  =  ( ( *  o.  F )  o.  * )   &    |-  (
 ( ph  /\  x  e.  S )  ->  ( * `  x )  e.  S )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   =>    |-  ( ph  ->  G  e.  (Poly `  S )
 )
 
Theoremcoecj 19491 Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  N  =  (deg `  F )   &    |-  G  =  ( ( *  o.  F )  o.  * )   &    |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  (coeff `  G )  =  ( *  o.  A ) )
 
Theoremplyrecj 19492 A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  ( * `  ( F `  A ) )  =  ( F `  ( * `  A ) ) )
 
Theoremplymul0or 19493 Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( ( F  o F  x.  G )  =  0 p 
 <->  ( F  =  0 p  \/  G  =  0 p ) ) )
 
Theoremofmulrt 19494 The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( `' ( F  o F  x.  G ) " {
 0 } )  =  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } )
 ) )
 
Theoremplyreres 19495 Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )
 
Theoremdvply1 19496* Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  G  =  ( z  e. 
 CC  |->  sum_ k  e.  (
 0 ... ( N  -  1 ) ) ( ( B `  k
 )  x.  ( z ^ k ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  B  =  ( k  e.  NN0  |->  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( CC  _D  F )  =  G )
 
Theoremdvply2g 19497 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.)
 |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) ) 
 ->  ( CC  _D  F )  e.  (Poly `  S ) )
 
Theoremdvply2 19498 The derivative of a polynomial is a polynomial. (Contributed by Stefan O'Rear, 14-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
 |-  ( F  e.  (Poly `  S )  ->  ( CC  _D  F )  e.  (Poly `  CC )
 )
 
Theoremdvnply2 19499 Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  ( ( CC  D n F ) `  N )  e.  (Poly `  S ) )
 
Theoremdvnply 19500 Polynomials have polynomials as derivatives of all orders. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
 |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  ( ( CC  D n F ) `  N )  e.  (Poly `  CC ) )
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