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Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdgrsub 19601 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 19602* 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 19603* Lemma for dgrco 19604. (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 19604 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 19605* Lemma for plycj 19606 and coecj 19607. (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 19606* 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 19607 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 19608 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 19609 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 19610 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 19611 Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )
 
Theoremdvply1 19612* 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 19613 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 19614 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 19615 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 19616 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 19617 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 19618 Extend class notation to include the quotient of a polynomial division.
 class quot
 
Definitiondf-quot 19619* 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 19620* 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 19621* Lemma for plydivalg 19627. (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 19622* Lemma for plydivalg 19627. (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 19623* Lemma for plydivex 19625. 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 19624* Lemma for plydivex 19625. 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 19625* Lemma for plydivalg 19627. (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 19626* Lemma for plydivalg 19627. (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 19627* 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 19628* 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 19629* 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 19630 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 19631 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 19632 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 19633 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12669). 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 19634 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 19635* Lemma for fta1 19636. (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 19636 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 19637 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 19638* Lemma for vieta1 19640. (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 19639* Lemma for vieta1 19640: 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 19640* 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 19641* 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 19642 Extend class notation to include the set of algebraic numbers.
 class  AA
 
Definitiondf-aa 19643 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 19644* 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 19645 An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  AA  ->  A  e.  CC )
 
Theoremaasscn 19646 The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |- 
 AA  C_  CC
 
Theoremelqaalem1 19647* Lemma for elqaa 19650. 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 19648* Lemma for elqaa 19650. (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 19649* Lemma for elqaa 19650. (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 19650* 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 19644 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 19651 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  QQ  ->  A  e.  AA )
 
Theoremqssaa 19652 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |- 
 QQ  C_  AA
 
Theoremiaa 19653 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  _i  e.  AA
 
Theoremaareccl 19654 The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( A  e.  AA  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  AA )
 
Theoremaacjcl 19655 The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( A  e.  AA  ->  ( * `  A )  e.  AA )
 
Theoremaannenlem1 19656* Lemma for aannen 19659. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  H  =  ( a  e.  NN0  |->  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p 
 /\  (deg `  d
 )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
 ) )  <_  a
 ) }  ( c `
  b )  =  0 } )   =>    |-  ( A  e.  NN0 
 ->  ( H `  A )  e.  Fin )
 
Theoremaannenlem2 19657* Lemma for aannen 19659. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  H  =  ( a  e.  NN0  |->  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p 
 /\  (deg `  d
 )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
 ) )  <_  a
 ) }  ( c `
  b )  =  0 } )   =>    |-  AA  =  U. ran  H
 
Theoremaannenlem3 19658* The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  H  =  ( a  e.  NN0  |->  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p 
 /\  (deg `  d
 )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
 ) )  <_  a
 ) }  ( c `
  b )  =  0 } )   =>    |-  AA  ~~  NN
 
Theoremaannen 19659 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |- 
 AA  ~~  NN
 
13.1.7  Liouville's approximation theorem
 
Theoremaalioulem1 19660 Lemma for aaliou 19666. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |-  ( ph  ->  F  e.  (Poly `  ZZ )
 )   &    |-  ( ph  ->  X  e.  ZZ )   &    |-  ( ph  ->  Y  e.  NN )   =>    |-  ( ph  ->  ( ( F `  ( X  /  Y ) )  x.  ( Y ^
 (deg `  F )
 ) )  e.  ZZ )
 
Theoremaalioulem2 19661* Lemma for aaliou 19666. (Contributed by Stefan O'Rear, 15-Nov-2014.)
 |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  ZZ ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  (
 ( F `  ( p  /  q ) )  =  0  ->  ( A  =  ( p  /  q )  \/  ( x  /  ( q ^ N ) )  <_  ( abs `  ( A  -  ( p  /  q
 ) ) ) ) ) )
 
Theoremaalioulem3 19662* Lemma for aaliou 19666. (Contributed by Stefan O'Rear, 15-Nov-2014.)
 |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  ZZ ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( F `  A )  =  0 )   =>    |-  ( ph  ->  E. x  e.  RR+  A. r  e.  RR  ( ( abs `  ( A  -  r ) ) 
 <_  1  ->  ( x  x.  ( abs `  ( F `  r ) ) )  <_  ( abs `  ( A  -  r
 ) ) ) )
 
Theoremaalioulem4 19663* Lemma for aaliou 19666. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  ZZ ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( F `  A )  =  0 )   =>    |-  ( ph  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  (
 ( ( F `  ( p  /  q
 ) )  =/=  0  /\  ( abs `  ( A  -  ( p  /  q ) ) ) 
 <_  1 )  ->  ( A  =  ( p  /  q )  \/  ( x  /  ( q ^ N ) )  <_  ( abs `  ( A  -  ( p  /  q
 ) ) ) ) ) )
 
Theoremaalioulem5 19664* Lemma for aaliou 19666. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  ZZ ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( F `  A )  =  0 )   =>    |-  ( ph  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  (
 ( F `  ( p  /  q ) )  =/=  0  ->  ( A  =  ( p  /  q )  \/  ( x  /  ( q ^ N ) )  <_  ( abs `  ( A  -  ( p  /  q
 ) ) ) ) ) )
 
Theoremaalioulem6 19665* Lemma for aaliou 19666. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  ZZ ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( F `  A )  =  0 )   =>    |-  ( ph  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  ( q ^ N ) )  <_  ( abs `  ( A  -  ( p  /  q
 ) ) ) ) )
 
Theoremaaliou 19666* Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial 
F in integer coefficients, is not approximable beyond order  N  = deg ( F ) by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  ZZ ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( F `  A )  =  0 )   =>    |-  ( ph  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  ( q ^ N ) )  < 
 ( abs `  ( A  -  ( p  /  q
 ) ) ) ) )
 
Theoremgeolim3 19667* Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( abs `  B )  <  1 )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )   =>    |-  ( ph  ->  seq  A (  +  ,  F ) 
 ~~>  ( C  /  (
 1  -  B ) ) )
 
Theoremaaliou2 19668* Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( A  e.  ( AA  i^i  RR )  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e. 
 NN  ( A  =  ( p  /  q
 )  \/  ( x 
 /  ( q ^
 k ) )  < 
 ( abs `  ( A  -  ( p  /  q
 ) ) ) ) )
 
Theoremaaliou2b 19669* Liouville's approximation theorem extended to complex  A. (Contributed by Stefan O'Rear, 20-Nov-2014.)
 |-  ( A  e.  AA  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e. 
 NN  ( A  =  ( p  /  q
 )  \/  ( x 
 /  ( q ^
 k ) )  < 
 ( abs `  ( A  -  ( p  /  q
 ) ) ) ) )
 
Theoremaaliou3lem1 19670* Lemma for aaliou3 19679. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  G  =  ( c  e.  ( ZZ>= `  A )  |->  ( ( 2 ^ -u ( ! `  A ) )  x.  ( ( 1  / 
 2 ) ^ (
 c  -  A ) ) ) )   =>    |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A )
 )  ->  ( G `  B )  e.  RR )
 
Theoremaaliou3lem2 19671* Lemma for aaliou3 19679. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  G  =  ( c  e.  ( ZZ>= `  A )  |->  ( ( 2 ^ -u ( ! `  A ) )  x.  ( ( 1  / 
 2 ) ^ (
 c  -  A ) ) ) )   &    |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a ) ) )   =>    |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>=
 `  A ) ) 
 ->  ( F `  B )  e.  ( 0 (,] ( G `  B ) ) )
 
Theoremaaliou3lem3 19672* Lemma for aaliou3 19679. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  G  =  ( c  e.  ( ZZ>= `  A )  |->  ( ( 2 ^ -u ( ! `  A ) )  x.  ( ( 1  / 
 2 ) ^ (
 c  -  A ) ) ) )   &    |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a ) ) )   =>    |-  ( A  e.  NN  ->  (  seq  A (  +  ,  F )  e.  dom  ~~>  /\  sum_ b  e.  ( ZZ>= `  A )
 ( F `  b
 )  e.  RR+  /\  sum_ b  e.  ( ZZ>= `  A ) ( F `  b )  <_  ( 2  x.  ( 2 ^ -u ( ! `  A ) ) ) ) )
 
Theoremaaliou3lem8 19673* Lemma for aaliou3 19679. (Contributed by Stefan O'Rear, 20-Nov-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  RR+ )  ->  E. x  e.  NN  ( 2  x.  (
 2 ^ -u ( ! `  ( x  +  1 ) ) ) )  <_  ( B  /  ( ( 2 ^
 ( ! `  x ) ) ^ A ) ) )
 
Theoremaaliou3lem4 19674* Lemma for aaliou3 19679. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a ) ) )   &    |-  L  =  sum_ b  e. 
 NN  ( F `  b )   &    |-  H  =  ( c  e.  NN  |->  sum_ b  e.  ( 1 ... c ) ( F `
  b ) )   =>    |-  L  e.  RR
 
Theoremaaliou3lem5 19675* Lemma for aaliou3 19679. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a ) ) )   &    |-  L  =  sum_ b  e. 
 NN  ( F `  b )   &    |-  H  =  ( c  e.  NN  |->  sum_ b  e.  ( 1 ... c ) ( F `
  b ) )   =>    |-  ( A  e.  NN  ->  ( H `  A )  e.  RR )
 
Theoremaaliou3lem6 19676* Lemma for aaliou3 19679. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a ) ) )   &    |-  L  =  sum_ b  e. 
 NN  ( F `  b )   &    |-  H  =  ( c  e.  NN  |->  sum_ b  e.  ( 1 ... c ) ( F `
  b ) )   =>    |-  ( A  e.  NN  ->  ( ( H `  A )  x.  (
 2 ^ ( ! `
  A ) ) )  e.  ZZ )
 
Theoremaaliou3lem7 19677* Lemma for aaliou3 19679. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a ) ) )   &    |-  L  =  sum_ b  e. 
 NN  ( F `  b )   &    |-  H  =  ( c  e.  NN  |->  sum_ b  e.  ( 1 ... c ) ( F `
  b ) )   =>    |-  ( A  e.  NN  ->  ( ( H `  A )  =/=  L  /\  ( abs `  ( L  -  ( H `  A ) ) )  <_  ( 2  x.  (
 2 ^ -u ( ! `  ( A  +  1 ) ) ) ) ) )
 
Theoremaaliou3lem9 19678* Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 20-Nov-2014.)
 |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a ) ) )   &    |-  L  =  sum_ b  e. 
 NN  ( F `  b )   &    |-  H  =  ( c  e.  NN  |->  sum_ b  e.  ( 1 ... c ) ( F `
  b ) )   =>    |-  -.  L  e.  AA
 
Theoremaaliou3 19679 Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 23-Nov-2014.)
 |- 
 sum_ k  e.  NN  ( 2 ^ -u ( ! `  k ) ) 
 e/  AA
 
13.2  Sequences and series
 
13.2.1  Taylor polynomials and Taylor's theorem
 
Syntaxctayl 19680 Taylor polynomial of a function.
 class Tayl
 
Syntaxcana 19681 The class of analytic functions.
 class Ana
 
Definitiondf-tayl 19682* Define the Taylor polynomial or Taylor series of a function. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |- Tayl  =  ( s  e.  { RR ,  CC } ,  f  e.  ( CC  ^pm  s )  |->  ( n  e.  ( NN0  u.  { 
 +oo } ) ,  a  e.  |^|_ k  e.  (
 ( 0 [,] n )  i^i  ZZ ) dom  ( ( s  D n f ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  (
 ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( s  D n f ) `  k ) `
  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k ) ) ) ) ) ) )
 
Definitiondf-ana 19683* Define the set of analytic functions, which are functions such that the Taylor series of the function at each point converges to the function in some neighborhood of the point. (Contributed by Mario Carneiro, 31-Dec-2016.)
 |- Ana 
 =  ( s  e. 
 { RR ,  CC } 
 |->  { f  e.  ( CC  ^pm  s )  | 
 A. x  e.  dom  f  x  e.  (
 ( int `  ( ( TopOpen ` fld )t  s ) ) `  dom  (  f  i^i  (  +oo ( s Tayl  f ) x ) ) ) } )
 
Theoremtaylfvallem1 19684* Lemma for taylfval 19686. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo ) )   &    |-  (
 ( ph  /\  k  e.  ( ( 0 [,]
 N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `  k
 ) )   =>    |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  (
 ( 0 [,] N )  i^i  ZZ ) ) 
 ->  ( ( ( ( ( S  D n F ) `  k
 ) `  B )  /  ( ! `  k
 ) )  x.  (
 ( X  -  B ) ^ k ) )  e.  CC )
 
Theoremtaylfvallem 19685* Lemma for taylfval 19686. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo ) )   &    |-  (
 ( ph  /\  k  e.  ( ( 0 [,]
 N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `  k
 ) )   =>    |-  ( ( ph  /\  X  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i 
 ZZ )  |->  ( ( ( ( ( S  D n F ) `
  k ) `  B )  /  ( ! `  k ) )  x.  ( ( X  -  B ) ^
 k ) ) ) )  C_  CC )
 
Theoremtaylfval 19686* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 
S is the base set with respect to evaluate the derivatives (generally  RR or 
CC),  F is the function we are approximating, at point  B, to order  N. The result is a polynomial function of  x.

This "extended" version of taylpfval 19692 additionally handles the case  N  =  +oo, in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)

 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo ) )   &    |-  (
 ( ph  /\  k  e.  ( ( 0 [,]
 N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `  k
 ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   =>    |-  ( ph  ->  T  =  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^
 k ) ) ) ) ) )
 
Theoremeltayl 19687* Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo ) )   &    |-  (
 ( ph  /\  k  e.  ( ( 0 [,]
 N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `  k
 ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   =>    |-  ( ph  ->  ( X T Y  <->  ( X  e.  CC  /\  Y  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i 
 ZZ )  |->  ( ( ( ( ( S  D n F ) `
  k ) `  B )  /  ( ! `  k ) )  x.  ( ( X  -  B ) ^
 k ) ) ) ) ) ) )
 
Theoremtaylf 19688* The Taylor series defines a function on a subset of the complexes. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo ) )   &    |-  (
 ( ph  /\  k  e.  ( ( 0 [,]
 N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `  k
 ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   =>    |-  ( ph  ->  T : dom  T --> CC )
 
Theoremtayl0 19689* The Taylor series is always defined at the basepoint, with value equal to the value of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo ) )   &    |-  (
 ( ph  /\  k  e.  ( ( 0 [,]
 N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `  k
 ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   =>    |-  ( ph  ->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) )
 
Theoremtaylplem1 19690* Lemma for taylpfval 19692 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  N ) )   =>    |-  ( ( ph  /\  k  e.  ( ( 0 [,]
 N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `  k
 ) )
 
Theoremtaylplem2 19691* Lemma for taylpfval 19692 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  N ) )   =>    |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  (
 0 ... N ) ) 
 ->  ( ( ( ( ( S  D n F ) `  k
 ) `  B )  /  ( ! `  k
 ) )  x.  (
 ( X  -  B ) ^ k ) )  e.  CC )
 
Theoremtaylpfval 19692* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 
S is the base set with respect to evaluate the derivatives (generally  RR or 
CC),  F is the function we are approximating, at point  B, to order  N. The result is a polynomial function of  x. (Contributed by Mario Carneiro, 31-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  N ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   =>    |-  ( ph  ->  T  =  ( x  e.  CC  |->  sum_
 k  e.  ( 0
 ... N ) ( ( ( ( ( S  D n F ) `  k ) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^
 k ) ) ) )
 
Theoremtaylpf 19693 The Taylor polynomial is a function on the complexes (even if the base set of the original function is the reals). (Contributed by Mario Carneiro, 31-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  N ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   =>    |-  ( ph  ->  T : CC --> CC )
 
Theoremtaylpval 19694* Value of the Taylor polynomial. (Contributed by Mario Carneiro, 31-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  N ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   &    |-  ( ph  ->  X  e.  CC )   =>    |-  ( ph  ->  ( T `  X )  =  sum_ k  e.  (
 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `  B )  /  ( ! `  k ) )  x.  ( ( X  -  B ) ^
 k ) ) )
 
Theoremtaylply2 19695* The Taylor polynomial is a polynomial of degree (at most)  N. This version of taylply 19696 shows that the coefficients of  T are in a subring of the complexes. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  N ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   &    |-  ( ph  ->  D  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  B  e.  D )   &    |-  (
 ( ph  /\  k  e.  ( 0 ... N ) )  ->  ( ( ( ( S  D n F ) `  k
 ) `  B )  /  ( ! `  k
 ) )  e.  D )   =>    |-  ( ph  ->  ( T  e.  (Poly `  D )  /\  (deg `  T )  <_  N ) )
 
Theoremtaylply 19696 The Taylor polynomial is a polynomial of degree (at most)  N. (Contributed by Mario Carneiro, 31-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  N ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   =>    |-  ( ph  ->  ( T  e.  (Poly `  CC )  /\  (deg `  T )  <_  N ) )
 
Theoremdvtaylp 19697 The derivative of the Taylor polynomial is the Taylor polynomial of the derivative of the function. (Contributed by Mario Carneiro, 31-Dec-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  ( N  +  1 ) ) )   =>    |-  ( ph  ->  ( CC  _D  ( ( N  +  1 ) ( S Tayl 
 F ) B ) )  =  ( N ( S Tayl  ( S  _D  F ) ) B ) )
 
Theoremdvntaylp 19698 The  M-th derivative of the Taylor polynomial is the Taylor polynomial of the  M-th derivative of the function. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  ( N  +  M ) ) )   =>    |-  ( ph  ->  ( ( CC  D n ( ( N  +  M ) ( S Tayl  F ) B ) ) `  M )  =  ( N ( S Tayl  (
 ( S  D n F ) `  M ) ) B ) )
 
Theoremdvntaylp0 19699 The first  N derivatives of the Taylor polynomial at  B match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  M  e.  ( 0 ... N ) )   &    |-  ( ph  ->  B  e.  dom  ( ( S  D n F ) `
  N ) )   &    |-  T  =  ( N ( S Tayl  F ) B )   =>    |-  ( ph  ->  (
 ( ( CC  D n T ) `  M ) `  B )  =  ( ( ( S  D n F ) `
  M ) `  B ) )
 
Theoremtaylthlem1 19700* Lemma for taylth 19702. This is the main part of Taylor's theorem, except for the induction step, which is supposed to be proven using L'Hôpital's rule. However, since our proof of L'Hôpital assumes that  S  =  RR, we can only do this part generically, and for taylth 19702 itself we must restrict to  RR. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  dom  (
 ( S  D n F ) `  N )  =  A )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  B  e.  A )   &    |-  T  =  ( N ( S Tayl  F ) B )   &    |-  R  =  ( x  e.  ( A 
 \  { B }
 )  |->  ( ( ( F `  x )  -  ( T `  x ) )  /  ( ( x  -  B ) ^ N ) ) )   &    |-  (
 ( ph  /\  ( n  e.  ( 1..^ N )  /\  0  e.  (
 ( y  e.  ( A  \  { B }
 )  |->  ( ( ( ( ( S  D n F ) `  ( N  -  n ) ) `
  y )  -  ( ( ( CC 
 D n T ) `
  ( N  -  n ) ) `  y ) )  /  ( ( y  -  B ) ^ n ) ) ) lim CC  B ) ) ) 
 ->  0  e.  (
 ( x  e.  ( A  \  { B }
 )  |->  ( ( ( ( ( S  D n F ) `  ( N  -  ( n  +  1 ) ) ) `
  x )  -  ( ( ( CC 
 D n T ) `
  ( N  -  ( n  +  1
 ) ) ) `  x ) )  /  ( ( x  -  B ) ^ ( n  +  1 )
 ) ) ) lim CC  B ) )   =>    |-  ( ph  ->  0  e.  ( R lim CC  B ) )
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