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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvply1 20201* 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 20202 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 20203 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 20204 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 20205 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 20206 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 20207 Extend class notation to include the quotient of a polynomial division.
 class quot
 
Definitiondf-quot 20208* 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 20209* 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 20210* Lemma for plydivalg 20216. (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 20211* Lemma for plydivalg 20216. (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 20212* Lemma for plydivex 20214. 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 20213* Lemma for plydivex 20214. 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 20214* Lemma for plydivalg 20216. (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 20215* Lemma for plydivalg 20216. (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 20216* 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 20217* 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 20218* 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 20219 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 20220 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 20221 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 20222 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 13042). 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 20223 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 20224* Lemma for fta1 20225. (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 20225 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 20226 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 20227* Lemma for vieta1 20229. (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 20228* Lemma for vieta1 20229: 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 20229* 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 20230* 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 20231 Extend class notation to include the set of algebraic numbers.
 class  AA
 
Definitiondf-aa 20232 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 20233* 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 20234 An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  AA  ->  A  e.  CC )
 
Theoremaasscn 20235 The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |- 
 AA  C_  CC
 
Theoremelqaalem1 20236* Lemma for elqaa 20239. 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 20237* Lemma for elqaa 20239. (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 20238* Lemma for elqaa 20239. (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 20239* 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 20233 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 20240 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  QQ  ->  A  e.  AA )
 
Theoremqssaa 20241 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |- 
 QQ  C_  AA
 
Theoremiaa 20242 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  _i  e.  AA
 
Theoremaareccl 20243 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 20244 The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( A  e.  AA  ->  ( * `  A )  e.  AA )
 
Theoremaannenlem1 20245* Lemma for aannen 20248. (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 20246* Lemma for aannen 20248. (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 20247* 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 20248 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |- 
 AA  ~~  NN
 
13.1.7  Liouville's approximation theorem
 
Theoremaalioulem1 20249 Lemma for aaliou 20255. 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 20250* Lemma for aaliou 20255. (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 20251* Lemma for aaliou 20255. (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 20252* Lemma for aaliou 20255. (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 20253* Lemma for aaliou 20255. (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 20254* Lemma for aaliou 20255. (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 20255* 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 20256* 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 20257* 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 20258* 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 20259* Lemma for aaliou3 20268. (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 20260* Lemma for aaliou3 20268. (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 20261* Lemma for aaliou3 20268. (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 20262* Lemma for aaliou3 20268. (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 20263* Lemma for aaliou3 20268. (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 20264* Lemma for aaliou3 20268. (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 20265* Lemma for aaliou3 20268. (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 20266* Lemma for aaliou3 20268. (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 20267* 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 20268 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 20269 Taylor polynomial of a function.
 class Tayl
 
Syntaxcana 20270 The class of analytic functions.
 class Ana
 
Definitiondf-tayl 20271* 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 20272* 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 20273* Lemma for taylfval 20275. (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 20274* Lemma for taylfval 20275. (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 20275* 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 20281 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 20276* 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 20277* 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 20278* 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 20279* Lemma for taylpfval 20281 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 20280* Lemma for taylpfval 20281 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 20281* 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 20282 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 20283* 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 20284* The Taylor polynomial is a polynomial of degree (at most)  N. This version of taylply 20285 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 20285 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 20286 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 20287 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 20288 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 20289* Lemma for taylth 20291. 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 20291 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 ) )
 
Theoremtaylthlem2 20290* Lemma for taylth 20291. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  dom  (
 ( RR  D n F ) `  N )  =  A )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  B  e.  A )   &    |-  T  =  ( N ( RR Tayl  F ) B )   &    |-  ( ph  ->  M  e.  ( 1..^ N ) )   &    |-  ( ph  ->  0  e.  ( ( x  e.  ( A  \  { B } )  |->  ( ( ( ( ( RR  D n F ) `  ( N  -  M ) ) `  x )  -  (
 ( ( CC  D n T ) `  ( N  -  M ) ) `
  x ) ) 
 /  ( ( x  -  B ) ^ M ) ) ) lim
 CC  B ) )   =>    |-  ( ph  ->  0  e.  ( ( x  e.  ( A  \  { B } )  |->  ( ( ( ( ( RR 
 D n F ) `
  ( N  -  ( M  +  1
 ) ) ) `  x )  -  (
 ( ( CC  D n T ) `  ( N  -  ( M  +  1 ) ) ) `
  x ) ) 
 /  ( ( x  -  B ) ^
 ( M  +  1 ) ) ) ) lim
 CC  B ) )
 
Theoremtaylth 20291* Taylor's theorem. The Taylor polynomial of a  N-times differentiable function is such that the error term goes to zero faster than  ( x  -  B ) ^ N. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  dom  (
 ( RR  D n F ) `  N )  =  A )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  B  e.  A )   &    |-  T  =  ( N ( RR Tayl  F ) B )   &    |-  R  =  ( x  e.  ( A 
 \  { B }
 )  |->  ( ( ( F `  x )  -  ( T `  x ) )  /  ( ( x  -  B ) ^ N ) ) )   =>    |-  ( ph  ->  0  e.  ( R lim CC  B ) )
 
13.2.2  Uniform convergence
 
Syntaxculm 20292 Extend class notation to include the uniform convergence predicate.
 class  ~~> u
 
Definitiondf-ulm 20293* Define the uniform convergence of a sequence of functions. Here  F ( ~~> u `  S ) G if  F is a sequence of functions  F ( n ) ,  n  e.  NN defined on  S and  G is a function on  S, and for every  0  <  x there is a  j such that the functions  F ( k ) for  j  <_  k are all uniformly within  x of  G on the domain  S. Compare with df-clim 12282. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ~~> u  =  ( s  e.  _V  |->  { <. f ,  y >.  |  E. n  e. 
 ZZ  ( f : ( ZZ>= `  n ) --> ( CC  ^m  s ) 
 /\  y : s --> CC  /\  A. x  e.  RR+  E. j  e.  ( ZZ>= `  n ) A. k  e.  ( ZZ>=
 `  j ) A. z  e.  s  ( abs `  ( ( ( f `  k ) `
  z )  -  ( y `  z
 ) ) )  < 
 x ) } )
 
Theoremulmrel 20294 The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |- 
 Rel  ( ~~> u `  S )
 
Theoremulmscl 20295 Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( F ( ~~> u `  S ) G  ->  S  e.  _V )
 
Theoremulmval 20296* Express the predicate: The sequence of functions  F converges uniformly to  G on  S. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( S  e.  V  ->  ( F ( ~~> u `  S ) G  <->  E. n  e.  ZZ  ( F : ( ZZ>= `  n ) --> ( CC 
 ^m  S )  /\  G : S --> CC  /\  A. x  e.  RR+  E. j  e.  ( ZZ>= `  n ) A. k  e.  ( ZZ>=
 `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k ) `
  z )  -  ( G `  z ) ) )  <  x ) ) )
 
Theoremulmcl 20297 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( F ( ~~> u `  S ) G  ->  G : S --> CC )
 
Theoremulmf 20298* Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( F ( ~~> u `  S ) G  ->  E. n  e.  ZZ  F : ( ZZ>= `  n )
 --> ( CC  ^m  S ) )
 
Theoremulmpm 20299 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( F ( ~~> u `  S ) G  ->  F  e.  ( ( CC 
 ^m  S )  ^pm  ZZ ) )
 
Theoremulmf2 20300 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G ) 
 ->  F : Z --> ( CC 
 ^m  S ) )
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