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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremig1pval2 20101 Generator of the zero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |- 
 .0.  =  ( 0g `  P )   =>    |-  ( R  e.  Ring  ->  ( G `  {  .0.  } )  =  .0.  )
 
Theoremig1pval3 20102 Characterizing properties of the monic generator of a nonzero ideal of polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  U  =  (LIdeal `  P )   &    |-  D  =  ( deg1  `  R )   &    |-  M  =  (Monic1p `  R )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
 )  ->  ( ( G `  I )  e.  I  /\  ( G `
  I )  e.  M  /\  ( D `
  ( G `  I ) )  = 
 sup ( ( D
 " ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
 
Theoremig1pcl 20103 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 20104 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 20105 Any ideal of polynomials over a division ring is generated by the ideal's canonical generator. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  G  =  (idlGen1p `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  K  =  (RSpan `  P )   =>    |-  ( ( R  e.  DivRing  /\  I  e.  U ) 
 ->  I  =  ( K `  { ( G `
  I ) }
 ) )
 
Theoremply1lpir 20106 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 20107 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 )
 
13.1.4  Elementary properties of complex polynomials
 
Syntaxcply 20108 Extend class notation to include the set of complex polynomials.
 class Poly
 
Syntaxcidp 20109 Extend class notation to include the identity polynomial.
 class  X p
 
Syntaxccoe 20110 Extend class notation to include the coefficient function on polynomials.
 class coeff
 
Syntaxcdgr 20111 Extend class notation to include the degree function on polynomials.
 class deg
 
Definitiondf-ply 20112* Define the set of polynomials on the complexes with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |- Poly  =  ( 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 20113 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  X p  =  (  _I  |`  CC )
 
Definitiondf-coe 20114* 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 20115 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 20116* 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 20117* 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 20118 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 20119* 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 20120* 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 20121 The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  (Poly `  ( S  u.  { 0 } )
 )  =  (Poly `  S )
 
Theoremplyf 20122 The polynomial is a function on the complexes. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F : CC --> CC )
 
Theoremplyss 20123 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 20124 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 20125* 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 20126* 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 20127* Lemma for ply1term 20128. (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 20128* 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 20129* 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 20130 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 20131 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 20132 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( S  C_  CC  ->  0 p  e.  (Poly `  S ) )
 
Theoremplyid 20133 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 20134* Lemma for plyeq0 20135. 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 20135* If a polynomial is zero at every point (or even just zero at the positive integers), then all the coefficients must be zero. This is the basis for the method of equating coefficients of equal polynomials, and ensures that df-coe 20114 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( 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 20136 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 20137* 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 20138* 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 20139* Lemma for plyadd 20141. (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 20140* Lemma for plymul 20142. (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 20141* 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 20142* 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 20143* 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 20144 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 20145 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 20146 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 20147* 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 20148* Lemma for coeeu 20149. (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 20149* 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 20150* 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 20151* 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 20152 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 20153* Lemma for dgrcl 20157 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 20154 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 20155 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 20156 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 20157 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 20158 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 20159 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 20160 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 20161* Lemma for coeid 20162. (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 20162* Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  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 20163* Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  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 20164* Reconstruct a polynomial as an explicit sum of the coefficient function up to at least the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  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 20165* 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 20166* 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 20167* Given an explicit expression for a polynomial, the degree is at most the highest term in the sum. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( 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 20168* If the highest term in a polynomial expression is nonzero, then the polynomial's degree is completely determined. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( 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 20169 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( A  e.  CC  ->  (deg `  ( CC  X. 
 { A } )
 )  =  0 )
 
Theorem0dgrb 20170 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 20171 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 20172 Lemma for coeadd 20174 and dgradd 20190. (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 20173* Lemma for coemul 20175 and dgrmul 20193. (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 20174 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 20175* 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 20176 The coefficient function is one-to-one, so if the coefficients are equal then the functions are equal and vice-versa. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S ) )  ->  ( F  =  G  <->  A  =  B ) )
 
Theoremcoemulhi 20177 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 20178 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 20179 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (coeff `  0 p
 )  =  ( NN0  X. 
 { 0 } )
 
Theoremcoesub 20180 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 20181* 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 20182* 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 20183* 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 20184 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F  e.  ( CC -cn-> CC )
 )
 
Theoremdgr0 20185 The degree of the zero polynomial is zero. Note: this differs from some other definitions of the degree of the zero polynomial, such as  -u 1 , 
-oo or undefined. But it is convenient for us to define it this way, so that we have dgrcl 20157, dgreq0 20188 and coeid 20162 without having to special-case zero, although plydivalg 20221 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (deg `  0 p
 )  =  0
 
Theoremcoeidp 20186 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 20187 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  (deg `  X p
 )  =  1
 
Theoremdgreq0 20188 The leading coefficient of a polynomial is nonzero, unless the entire polynomial is zero. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Fan Zheng, 21-Jun-2016.)
 |-  N  =  (deg `  F )   &    |-  A  =  (coeff `  F )   =>    |-  ( F  e.  (Poly `  S )  ->  ( F  =  0 p  <->  ( A `  N )  =  0 ) )
 
Theoremdgrlt 20189 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 20190 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 20191 The degree of a sum of polynomials of unequal degrees is the degree of the larger polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G ) )  =  N )
 
Theoremdgrmul2 20192 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 20193 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 20194 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 20195 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 20196* 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 20197* Lemma for dgrco 20198. (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 20198 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 20199* Lemma for plycj 20200 and coecj 20201. (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 20200* The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on  ( * `  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 )
 )
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