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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremig1pcl 23001 The monic generator of an ideal is always in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |-  U  =  (LIdeal `  P )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U ) 
 ->  ( G `  I
 )  e.  I )
 
Theoremig1pdvds 23002 The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  .||  =  ( ||r
 `  P )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I
 )  ->  ( G `  I )  .||  X )
 
Theoremig1prsp 23003 Any ideal of polynomials over a division ring is generated by the ideal's canonical generator. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  K  =  (RSpan `  P )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U ) 
 ->  I  =  ( K `  { ( G `
  I ) }
 ) )
 
Theoremply1lpir 23004 The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  DivRing  ->  P  e. LPIR )
 
Theoremply1pid 23005 The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e. Field  ->  P  e. PID )
 
14.1.3  Elementary properties of complex polynomials
 
Syntaxcply 23006 Extend class notation to include the set of complex polynomials.
 class Poly
 
Syntaxcidp 23007 Extend class notation to include the identity polynomial.
 class  Xp
 
Syntaxccoe 23008 Extend class notation to include the coefficient function on polynomials.
 class coeff
 
Syntaxcdgr 23009 Extend class notation to include the degree function on polynomials.
 class deg
 
Definitiondf-ply 23010* Define the set of polynomials on the complex numbers 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 23011 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  Xp  =  (  _I  |`  CC )
 
Definitiondf-coe 23012* 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 23013 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 23014* 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 23015* 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 23016 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 23017* 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 23018* 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 23019 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 23020 The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F : CC --> CC )
 
Theoremplyss 23021 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 23022 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 23023* 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 23024* 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 23025* Lemma for ply1term 23026. (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 23026* 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 23027* 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 23028 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 23029 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ( A  e.  CC  /\  ( F `  A )  =/=  0
 )  ->  F  =/=  0p )
 
Theoremply0 23030 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( S  C_  CC  ->  0p  e.  (Poly `  S ) )
 
Theoremplyid 23031 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( S  C_  CC  /\  1  e.  S )  ->  Xp  e.  (Poly `  S )
 )
 
Theoremplyeq0lem 23032* Lemma for plyeq0 23033. 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  ->  0p  =  ( 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 23033* 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 23012 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  ->  0p  =  ( z  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( A `  k
 )  x.  ( z ^ k ) ) ) )   =>    |-  ( ph  ->  A  =  ( NN0  X.  {
 0 } ) )
 
Theoremplypf1 23034 Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.) (Proof shortened by AV, 29-Sep-2019.)
 |-  R  =  (flds  S )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  (
 Base `  P )   &    |-  E  =  (eval1 ` fld )   =>    |-  ( S  e.  (SubRing ` fld ) 
 ->  (Poly `  S )  =  ( E " A ) )
 
Theoremplyaddlem1 23035* 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  oF  +  G )  =  ( z  e.  CC  |->  sum_ k  e.  (
 0 ... if ( M 
 <_  N ,  N ,  M ) ) ( ( ( A  oF  +  B ) `  k )  x.  (
 z ^ k ) ) ) )
 
Theoremplymullem1 23036* 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  oF  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 23037* Lemma for plyadd 23039. (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  oF  +  G )  e.  (Poly `  S ) )
 
Theoremplymullem 23038* Lemma for plymul 23040. (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  oF  x.  G )  e.  (Poly `  S ) )
 
Theoremplyadd 23039* 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  oF  +  G )  e.  (Poly `  S ) )
 
Theoremplymul 23040* 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  oF  x.  G )  e.  (Poly `  S ) )
 
Theoremplysub 23041* 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  oF  -  G )  e.  (Poly `  S ) )
 
Theoremplyaddcl 23042 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  oF  +  G )  e.  (Poly `  CC ) )
 
Theoremplymulcl 23043 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  oF  x.  G )  e.  (Poly `  CC ) )
 
Theoremplysubcl 23044 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  oF  -  G )  e.  (Poly `  CC ) )
 
Theoremcoeval 23045* 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 23046* Lemma for coeeu 23047. (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 23047* 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 23048* 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 23049* 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 23050 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 23051* Lemma for dgrcl 23055 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 23052 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 23053 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 23054 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 23055 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 23056 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 23057 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 23058 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 23059* Lemma for coeid 23060. (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 23060* 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 23061* 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 23062* 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 23063* 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 23064* 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 23065* 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 23066* 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 23067 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( A  e.  CC  ->  (deg `  ( CC  X. 
 { A } )
 )  =  0 )
 
Theorem0dgrb 23068 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 ) }
 ) ) )
 
Theoremdgrnznn 23069 A nonzero polynomial with a root has positive degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A )  =  0 ) )  ->  (deg `  P )  e. 
 NN )
 
Theoremcoefv0 23070 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 23071 Lemma for coeadd 23073 and dgradd 23089. (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  oF  +  G ) )  =  ( A  oF  +  B )  /\  (deg `  ( F  oF  +  G ) )  <_  if ( M  <_  N ,  N ,  M ) ) )
 
Theoremcoemullem 23072* Lemma for coemul 23074 and dgrmul 23092. (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  oF  x.  G ) )  =  ( n  e.  NN0  |->  sum_
 k  e.  ( 0
 ... n ) ( ( A `  k
 )  x.  ( B `
  ( n  -  k ) ) ) )  /\  (deg `  ( F  oF  x.  G ) )  <_  ( M  +  N ) ) )
 
Theoremcoeadd 23073 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  oF  +  G ) )  =  ( A  oF  +  B ) )
 
Theoremcoemul 23074* 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  oF  x.  G ) ) `  N )  =  sum_ k  e.  ( 0 ...
 N ) ( ( A `  k )  x.  ( B `  ( N  -  k
 ) ) ) )
 
Theoremcoe11 23075 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 23076 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  oF  x.  G ) ) `  ( M  +  N ) )  =  (
 ( A `  M )  x.  ( B `  N ) ) )
 
Theoremcoemulc 23077 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 } )  oF  x.  F ) )  =  (
 ( NN0  X.  { A } )  oF  x.  (coeff `  F )
 ) )
 
Theoremcoe0 23078 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (coeff `  0p
 )  =  ( NN0  X. 
 { 0 } )
 
Theoremcoesub 23079 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  oF  -  G ) )  =  ( A  oF  -  B ) )
 
Theoremcoe1termlem 23080* 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 23081* 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 23082* 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 23083 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F  e.  ( CC -cn-> CC )
 )
 
Theoremdgr0 23084 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 23055, dgreq0 23087 and coeid 23060 without having to special-case zero, although plydivalg 23120 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (deg `  0p
 )  =  0
 
Theoremcoeidp 23085 The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( A  e.  NN0  ->  ( (coeff `  Xp
 ) `  A )  =  if ( A  =  1 ,  1 , 
 0 ) )
 
Theoremdgrid 23086 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  (deg `  Xp
 )  =  1
 
Theoremdgreq0 23087 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  =  0p  <->  ( A `  N )  =  0 ) )
 
Theoremdgrlt 23088 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  =  0p  \/  N  <  M )  <->  ( N  <_  M 
 /\  ( A `  M )  =  0
 ) ) )
 
Theoremdgradd 23089 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  oF  +  G ) )  <_  if ( M  <_  N ,  N ,  M ) )
 
Theoremdgradd2 23090 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  oF  +  G ) )  =  N )
 
Theoremdgrmul2 23091 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  oF  x.  G ) )  <_  ( M  +  N ) )
 
Theoremdgrmul 23092 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  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( F  oF  x.  G )
 )  =  ( M  +  N ) )
 
Theoremdgrmulc 23093 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 } )  oF  x.  F ) )  =  (deg `  F ) )
 
Theoremdgrsub 23094 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  oF  -  G ) )  <_  if ( M  <_  N ,  N ,  M ) )
 
Theoremdgrcolem1 23095* 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 23096* Lemma for dgrco 23097. (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 23097 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 23098* Lemma for plycj 23099 and coecj 23100. (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 23099* 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 23100 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 ) )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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