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Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcoeeu 19601* 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 19602* 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 19603* 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 19604 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 19605* Lemma for dgrcl 19609 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 19606 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 19607 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 19608 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 19609 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 19610 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 19611 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 19612 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 19613* Lemma for coeid 19614. (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 19614* 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 19615* 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 19616* 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 19617* 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 19618* 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 19619* 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 19620* 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 19621 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( A  e.  CC  ->  (deg `  ( CC  X. 
 { A } )
 )  =  0 )
 
Theorem0dgrb 19622 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 19623 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 19624 Lemma for coeadd 19626 and dgradd 19642. (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 19625* Lemma for coemul 19627 and dgrmul 19645. (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 19626 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 19627* 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 19628 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 19629 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 19630 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 19631 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (coeff `  0 p
 )  =  ( NN0  X. 
 { 0 } )
 
Theoremcoesub 19632 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 19633* 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 19634* 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 19635* 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 19636 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( F  e.  (Poly `  S )  ->  F  e.  ( CC -cn-> CC )
 )
 
Theoremdgr0 19637 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 19609, dgreq0 19640 and coeid 19614 without having to special-case zero, although plydivalg 19673 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  (deg `  0 p
 )  =  0
 
Theoremcoeidp 19638 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 19639 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  (deg `  X p
 )  =  1
 
Theoremdgreq0 19640 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 19641 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 19642 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 19643 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 19644 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 19645 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 19646 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 19647 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 19648* 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 19649* Lemma for dgrco 19650. (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 19650 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 19651* Lemma for plycj 19652 and coecj 19653. (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 19652* 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 19653 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 19654 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 19655 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 19656 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 19657 Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )
 
Theoremdvply1 19658* 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 19659 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 19660 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 19661 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 19662 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 19663 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 19664 Extend class notation to include the quotient of a polynomial division.
 class quot
 
Definitiondf-quot 19665* 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 19666* 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 19667* Lemma for plydivalg 19673. (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 )
 
Theoremplydivlem2 19668* Lemma for plydivalg 19673. (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  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  =/=  0 p )   &    |-  R  =  ( F  o F  -  ( G  o F  x.  q ) )   =>    |-  ( ( ph  /\  q  e.  (Poly `  S ) )  ->  R  e.  (Poly `  S ) )
 
Theoremplydivlem3 19669* Lemma for plydivex 19671. Base case of induction. (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  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  =/=  0 p )   &    |-  R  =  ( F  o F  -  ( G  o F  x.  q ) )   &    |-  ( ph  ->  ( F  =  0 p  \/  (
 (deg `  F )  -  (deg `  G )
 )  <  0 )
 )   =>    |-  ( ph  ->  E. q  e.  (Poly `  S )
 ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )
 
Theoremplydivlem4 19670* Lemma for plydivex 19671. Induction step. (Contributed by Mario Carneiro, 26-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  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  =/=  0 p )   &    |-  R  =  ( F  o F  -  ( G  o F  x.  q ) )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  ( M  -  N )  =  D )   &    |-  ( ph  ->  F  =/=  0 p )   &    |-  U  =  ( f  o F  -  ( G  o F  x.  p ) )   &    |-  H  =  ( z  e.  CC  |->  ( ( ( A `  M )  /  ( B `  N ) )  x.  ( z ^ D ) ) )   &    |-  ( ph  ->  A. f  e.  (Poly `  S )
 ( ( f  =  0 p  \/  (
 (deg `  f )  -  N )  <  D )  ->  E. p  e.  (Poly `  S ) ( U  =  0 p  \/  (deg `  U )  <  N ) ) )   &    |-  A  =  (coeff `  F )   &    |-  B  =  (coeff `  G )   &    |-  M  =  (deg `  F )   &    |-  N  =  (deg `  G )   =>    |-  ( ph  ->  E. q  e.  (Poly `  S )
 ( R  =  0 p  \/  (deg `  R )  <  N ) )
 
Theoremplydivex 19671* Lemma for plydivalg 19673. (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  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  =/=  0 p )   &    |-  R  =  ( F  o F  -  ( G  o F  x.  q ) )   =>    |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
 (deg `  G )
 ) )
 
Theoremplydiveu 19672* Lemma for plydivalg 19673. (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  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  =/=  0 p )   &    |-  R  =  ( F  o F  -  ( G  o F  x.  q ) )   &    |-  ( ph  ->  q  e.  (Poly `  S ) )   &    |-  ( ph  ->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )   &    |-  T  =  ( F  o F  -  ( G  o F  x.  p ) )   &    |-  ( ph  ->  p  e.  (Poly `  S ) )   &    |-  ( ph  ->  ( T  =  0 p  \/  (deg `  T )  <  (deg `  G ) ) )   =>    |-  ( ph  ->  p  =  q )
 
Theoremplydivalg 19673* The division algorithm on polynomials over a subfield  S of the complex numbers. If  F and  G  =/=  0 are polynomials over  S, then there is a unique quotient polynomial  q such that the remainder  F  -  G  x.  q is either zero or has degree less than  G. (Contributed by Mario Carneiro, 26-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  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  =/=  0 p )   &    |-  R  =  ( F  o F  -  ( G  o F  x.  q ) )   =>    |-  ( ph  ->  E! q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
 (deg `  G )
 ) )
 
Theoremquotlem 19674* Lemma for properties of the polynomial quotient function. (Contributed by Mario Carneiro, 26-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  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  =/=  0 p )   &    |-  R  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )   =>    |-  ( ph  ->  (
 ( F quot  G )  e.  (Poly `  S )  /\  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) ) )
 
Theoremquotcl 19675* The quotient of two polynomials in a field  S is also in the field. (Contributed by Mario Carneiro, 26-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  ->  F  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  e.  (Poly `  S )
 )   &    |-  ( ph  ->  G  =/=  0 p )   =>    |-  ( ph  ->  ( F quot  G )  e.  (Poly `  S )
 )
 
Theoremquotcl2 19676 Closure of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F quot 
 G )  e.  (Poly `  CC ) )
 
Theoremquotdgr 19677 Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  R  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( R  =  0 p  \/  (deg `  R )  < 
 (deg `  G )
 ) )
 
Theoremplyremlem 19678 Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )   =>    |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G )  =  1  /\  ( `' G " { 0 } )  =  { A } ) )
 
Theoremplyrem 19679 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12715). If a polynomial  F is divided by the linear factor  x  -  A, the remainder is equal to  F ( A ), the evaluation of the polynomial at  A (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )   &    |-  R  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )   =>    |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC 
 X.  { ( F `  A ) } )
 )
 
Theoremfacth 19680 The factor theorem. If a polynomial  F has a root at 
A, then  G  =  x  -  A is a factor of  F (and the other factor is  F quot  G). (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )   =>    |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `  A )  =  0 )  ->  F  =  ( G  o F  x.  ( F quot  G ) ) )
 
Theoremfta1lem 19681* Lemma for fta1 19682. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  R  =  ( `' F " { 0 } )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  F  e.  ( (Poly `  CC )  \  { 0 p } ) )   &    |-  ( ph  ->  (deg `  F )  =  ( D  +  1 ) )   &    |-  ( ph  ->  A  e.  ( `' F " { 0 } ) )   &    |-  ( ph  ->  A. g  e.  (
 (Poly `  CC )  \  { 0 p }
 ) ( (deg `  g )  =  D  ->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " { 0 } ) )  <_  (deg `  g ) ) ) )   =>    |-  ( ph  ->  ( R  e.  Fin  /\  ( # `
  R )  <_  (deg `  F ) ) )
 
Theoremfta1 19682 The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg ( F ) roots. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  R  =  ( `' F " { 0 } )   =>    |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p ) 
 ->  ( R  e.  Fin  /\  ( # `  R )  <_  (deg `  F ) ) )
 
Theoremquotcan 19683 Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  H  =  ( F  o F  x.  G )   =>    |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H quot 
 G )  =  F )
 
Theoremvieta1lem1 19684* Lemma for vieta1 19686. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  R  =  ( `' F " { 0 } )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  ( # `  R )  =  N )   &    |-  ( ph  ->  D  e.  NN )   &    |-  ( ph  ->  ( D  +  1 )  =  N )   &    |-  ( ph  ->  A. f  e.  (Poly `  CC )
 ( ( D  =  (deg `  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
 )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u ( ( (coeff `  f ) `  ( (deg `  f
 )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) )   &    |-  Q  =  ( F quot  ( X p  o F  -  ( CC  X.  { z }
 ) ) )   =>    |-  ( ( ph  /\  z  e.  R ) 
 ->  ( Q  e.  (Poly `  CC )  /\  D  =  (deg `  Q )
 ) )
 
Theoremvieta1lem2 19685* Lemma for vieta1 19686: inductive step. Let  z be a root of  F. Then  F  =  ( X p  -  z
)  x.  Q for some  Q by the factor theorem, and  Q is a degree-  D polynomial, so by the induction hypothesis  sum_ x  e.  ( `' Q "
0 ) x  = 
-u (coeff `  Q
) `  ( D  -  1 )  /  (coeff `  Q
) `  D, so  sum_ x  e.  R x  =  z  -  (coeff `  Q
) `  ( D  - 
1 )  /  (coeff `  Q ) `  D. Now the coefficients of  F are  A `  ( D  +  1 )  =  (coeff `  Q
) `  D and  A `  D  =  sum_ k  e.  ( 0 ... D
) (coeff `  X p  -  z ) `  k  x.  (coeff `  Q )  `  ( D  -  k ), which works out to  -u z  x.  (coeff `  Q ) `  D  +  (coeff `  Q ) `  ( D  -  1 ), so putting it all together we have  sum_ x  e.  R x  =  -u A `  D  /  A `  ( D  +  1 ) as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  R  =  ( `' F " { 0 } )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  ( # `  R )  =  N )   &    |-  ( ph  ->  D  e.  NN )   &    |-  ( ph  ->  ( D  +  1 )  =  N )   &    |-  ( ph  ->  A. f  e.  (Poly `  CC )
 ( ( D  =  (deg `  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
 )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u ( ( (coeff `  f ) `  ( (deg `  f
 )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) )   &    |-  Q  =  ( F quot  ( X p  o F  -  ( CC  X.  { z }
 ) ) )   =>    |-  ( ph  ->  sum_
 x  e.  R  x  =  -u ( ( A `
  ( N  -  1 ) )  /  ( A `  N ) ) )
 
Theoremvieta1 19686* The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree  N has  N distinct roots, then the sum over these roots can be calculated as  -u A ( N  -  1 )  /  A ( N ). (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  R  =  ( `' F " { 0 } )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  ( # `  R )  =  N )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  sum_
 x  e.  R  x  =  -u ( ( A `
  ( N  -  1 ) )  /  ( A `  N ) ) )
 
Theoremplyexmo 19687* An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)
 |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  E* p ( p  e.  (Poly `  S )  /\  ( p  |`  D )  =  F ) )
 
13.1.6  Algebraic numbers
 
Syntaxcaa 19688 Extend class notation to include the set of algebraic numbers.
 class  AA
 
Definitiondf-aa 19689 Define the set of algebraic numbers. An algebraic number is a root of a nonzero polynomial over the integers. Here we construct it as the union of all kernels (preimages of 
{ 0 }) of all polynomials in  (Poly `  ZZ ), except the zero polynomial  0 p. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |- 
 AA  =  U_ f  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( `' f " { 0 } )
 
Theoremelaa 19690* Elementhood in the set of algebraic numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  {
 0 p } )
 ( f `  A )  =  0 )
 )
 
Theoremaacn 19691 An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  AA  ->  A  e.  CC )
 
Theoremaasscn 19692 The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |- 
 AA  C_  CC
 
Theoremelqaalem1 19693* Lemma for elqaa 19696. The function  N represents the denominators of the rational coefficients 
B. By multiplying them all together to make  R, we get a number big enough to clear all the denominators and make  R  x.  F an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  { 0 p } ) )   &    |-  ( ph  ->  ( F `  A )  =  0 )   &    |-  B  =  (coeff `  F )   &    |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )
 )   &    |-  R  =  (  seq  0 (  x.  ,  N ) `  (deg `  F ) )   =>    |-  ( ( ph  /\  K  e.  NN0 )  ->  (
 ( N `  K )  e.  NN  /\  (
 ( B `  K )  x.  ( N `  K ) )  e. 
 ZZ ) )
 
Theoremelqaalem2 19694* Lemma for elqaa 19696. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  { 0 p } ) )   &    |-  ( ph  ->  ( F `  A )  =  0 )   &    |-  B  =  (coeff `  F )   &    |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )
 )   &    |-  R  =  (  seq  0 (  x.  ,  N ) `  (deg `  F ) )   &    |-  P  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  x.  y ) 
 mod  ( N `  K ) ) )   =>    |-  ( ( ph  /\  K  e.  ( 0 ... (deg `  F ) ) ) 
 ->  ( R  mod  ( N `  K ) )  =  0 )
 
Theoremelqaalem3 19695* Lemma for elqaa 19696. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  { 0 p } ) )   &    |-  ( ph  ->  ( F `  A )  =  0 )   &    |-  B  =  (coeff `  F )   &    |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )
 )   &    |-  R  =  (  seq  0 (  x.  ,  N ) `  (deg `  F ) )   =>    |-  ( ph  ->  A  e.  AA )
 
Theoremelqaa 19696* The set of numbers generated by the roots of polynomials in the rational numbers is the same as the set of algebraic numbers, which by elaa 19690 are defined only in terms of polynomials over the integers. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  {
 0 p } )
 ( f `  A )  =  0 )
 )
 
Theoremqaa 19697 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  QQ  ->  A  e.  AA )
 
Theoremqssaa 19698 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |- 
 QQ  C_  AA
 
Theoremiaa 19699 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  _i  e.  AA
 
Theoremaareccl 19700 The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( A  e.  AA  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  AA )
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