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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremplyco0 19501* 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 19502* 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 19503 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 19504* 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 19505* 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 19506 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 19507 The polynomial is a function on the complexes. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F : CC --> CC )
 
Theoremplyss 19508 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 19509 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 19510* 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 19511* 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 19512* Lemma for ply1term 19513. (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 19513* 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 19514* 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 19515 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 19516 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 19517 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( S  C_  CC  ->  0 p  e.  (Poly `  S ) )
 
Theoremplyid 19518 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 19519* Lemma for plyeq0 19520. 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 19520* 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 19499 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 19521 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 19522* 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 19523* 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 19524* Lemma for plyadd 19526. (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 19525* Lemma for plymul 19527. (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 19526* 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 19527* 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 19528* 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 19529 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 19530 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 19531 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 19532* 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 19533* Lemma for coeeu 19534. (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 19534* 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 19535* 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 19536* 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 19537 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 19538* Lemma for dgrcl 19542 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 19539 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 19540 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 19541 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 19542 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 19543 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 19544 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 19545 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 19546* Lemma for coeid 19547. (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 19547* 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 19548* 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 19549* 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 19550* 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 19551* 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 19552* 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 19553* 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 19554 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( A  e.  CC  ->  (deg `  ( CC  X. 
 { A } )
 )  =  0 )
 
Theorem0dgrb 19555 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 19556 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 19557 Lemma for coeadd 19559 and dgradd 19575. (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 19558* Lemma for coemul 19560 and dgrmul 19578. (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 19559 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 19560* 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 19561 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 19562 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 19563 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 19564 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (coeff `  0 p
 )  =  ( NN0  X. 
 { 0 } )
 
Theoremcoesub 19565 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 19566* 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 19567* 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 19568* 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 19569 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F  e.  ( CC -cn-> CC )
 )
 
Theoremdgr0 19570 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 19542, dgreq0 19573 and coeid 19547 without having to special-case zero, although plydivalg 19606 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (deg `  0 p
 )  =  0
 
Theoremcoeidp 19571 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 19572 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  (deg `  X p
 )  =  1
 
Theoremdgreq0 19573 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 19574 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 19575 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 19576 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 19577 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 19578 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 19579 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 19580 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 19581* 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 19582* Lemma for dgrco 19583. (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 19583 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 19584* Lemma for plycj 19585 and coecj 19586. (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 19585* 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 19586 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 19587 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 19588 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 19589 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 19590 Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )
 
Theoremdvply1 19591* 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 19592 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 19593 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 19594 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 19595 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 ) )
 
Theoremplycpn 19596 Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( F  e.  (Poly `  S )  ->  F  e.  |^| ran  ( C ^n `  CC ) )
 
13.1.5  The Division algorithm for polynomials
 
Syntaxcquot 19597 Extend class notation to include the quotient of a polynomial division.
 class quot
 
Definitiondf-quot 19598* Define the quotient function on polynomials. This is the  q of the expression  f  =  g  x.  q  +  r in the division algorithm. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |- quot  =  ( f  e.  (Poly `  CC ) ,  g  e.  ( (Poly `  CC )  \  { 0 p } )  |->  ( iota_ q  e.  (Poly `  CC ) [. ( f  o F  -  ( g  o F  x.  q
 ) )  /  r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) ) ) )
 
Theoremquotval 19599* Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  R  =  ( F  o F  -  ( G  o F  x.  q
 ) )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F quot 
 G )  =  (
 iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  < 
 (deg `  G )
 ) ) )
 
Theoremplydivlem1 19600* Lemma for plydivalg 19606. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  +  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  x.  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  x  =/=  0 ) ) 
 ->  ( 1  /  x )  e.  S )   &    |-  ( ph  ->  -u 1  e.  S )   =>    |-  ( ph  ->  0  e.  S )
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