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Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelqaalem3 19701* Lemma for elqaa 19702. (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 19702* 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 19696 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 19703 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  ( A  e.  QQ  ->  A  e.  AA )
 
Theoremqssaa 19704 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |- 
 QQ  C_  AA
 
Theoremiaa 19705 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
 |-  _i  e.  AA
 
Theoremaareccl 19706 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 19707 The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( A  e.  AA  ->  ( * `  A )  e.  AA )
 
Theoremaannenlem1 19708* Lemma for aannen 19711. (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 19709* Lemma for aannen 19711. (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 19710* 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 19711 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |- 
 AA  ~~  NN
 
13.1.7  Liouville's approximation theorem
 
Theoremaalioulem1 19712 Lemma for aaliou 19718. 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 19713* Lemma for aaliou 19718. (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 19714* Lemma for aaliou 19718. (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 19715* Lemma for aaliou 19718. (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 19716* Lemma for aaliou 19718. (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 19717* Lemma for aaliou 19718. (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 19718* 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 19719* 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 19720* 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 19721* 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 19722* Lemma for aaliou3 19731. (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 19723* Lemma for aaliou3 19731. (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 19724* Lemma for aaliou3 19731. (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 19725* Lemma for aaliou3 19731. (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 19726* Lemma for aaliou3 19731. (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 19727* Lemma for aaliou3 19731. (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 19728* Lemma for aaliou3 19731. (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 19729* Lemma for aaliou3 19731. (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 19730* 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 19731 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 19732 Taylor polynomial of a function.
 class Tayl
 
Syntaxcana 19733 The class of analytic functions.
 class Ana
 
Definitiondf-tayl 19734* 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 19735* 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 19736* Lemma for taylfval 19738. (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 19737* Lemma for taylfval 19738. (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 19738* 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 19744 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 19739* 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 19740* 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 19741* 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 19742* Lemma for taylpfval 19744 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 19743* Lemma for taylpfval 19744 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 19744* 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 19745 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 19746* 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 19747* The Taylor polynomial is a polynomial of degree (at most)  N. This version of taylply 19748 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 19748 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 19749 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 19750 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 19751 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 19752* Lemma for taylth 19754. 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 19754 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 19753* Lemma for taylth 19754. (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 19754* 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 19755 Extend class notation to include the uniform convergence predicate.
 class  ~~> u
 
Definitiondf-ulm 19756* 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 11962. (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 19757 The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |- 
 Rel  ( ~~> u `  S )
 
Theoremulmscl 19758 Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( F ( ~~> u `  S ) G  ->  S  e.  _V )
 
Theoremulmval 19759* 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 19760 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( F ( ~~> u `  S ) G  ->  G : S --> CC )
 
Theoremulmf 19761* 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 19762 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 19763 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 ) )
 
Theoremulm2 19764* Simplify ulmval 19759 when  F and  G are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   &    |-  ( ( ph  /\  ( k  e.  Z  /\  z  e.  S ) )  ->  ( ( F `  k ) `
  z )  =  B )   &    |-  ( ( ph  /\  z  e.  S ) 
 ->  ( G `  z
 )  =  A )   &    |-  ( ph  ->  G : S
 --> CC )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  <  x ) )
 
Theoremulmi 19765* The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   &    |-  ( ( ph  /\  ( k  e.  Z  /\  z  e.  S ) )  ->  ( ( F `  k ) `
  z )  =  B )   &    |-  ( ( ph  /\  z  e.  S ) 
 ->  ( G `  z
 )  =  A )   &    |-  ( ph  ->  F ( ~~> u `  S ) G )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  <  C )
 
Theoremulmclm 19766* A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  H  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  (
 ( F `  k
 ) `  A )  =  ( H `  k
 ) )   &    |-  ( ph  ->  F ( ~~> u `  S ) G )   =>    |-  ( ph  ->  H  ~~>  ( G `  A ) )
 
Theoremulmres 19767 A sequence of functions converges iff the tail of the sequence converges (for any finite cutoff). (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   =>    |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  ( F  |`  W ) ( ~~> u `  S ) G ) )
 
Theoremulmshftlem 19768* Lemma for ulmshft 19769. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  ( M  +  K ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC 
 ^m  S ) )   &    |-  ( ph  ->  H  =  ( n  e.  W  |->  ( F `  ( n  -  K ) ) ) )   =>    |-  ( ph  ->  ( F ( ~~> u `  S ) G  ->  H ( ~~> u `  S ) G ) )
 
Theoremulmshft 19769* A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  ( M  +  K ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC 
 ^m  S ) )   &    |-  ( ph  ->  H  =  ( n  e.  W  |->  ( F `  ( n  -  K ) ) ) )   =>    |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  H ( ~~> u `  S ) G ) )
 
Theoremulm0 19770 Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   &    |-  ( ph  ->  G : S --> CC )   =>    |-  (
 ( ph  /\  S  =  (/) )  ->  F ( ~~> u `  S ) G )
 
Theoremulmcaulem 19771* Lemma for ulmcau 19772 and ulmcau2 19773: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 11839. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   =>    |-  ( ph  ->  (
 A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  (
 ( ( F `  k ) `  z
 )  -  ( ( F `  j ) `
  z ) ) )  <  x  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. m  e.  ( ZZ>= `  k ) A. z  e.  S  ( abs `  ( (
 ( F `  k
 ) `  z )  -  ( ( F `  m ) `  z
 ) ) )  < 
 x ) )
 
Theoremulmcau 19772* A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 
0  <  x there is a  j such that for all  j  <_  k the functions  F ( k ) and  F
( j ) are uniformly within  x of each other on  S. This is the four-quantifier version, see ulmcau2 19773 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   =>    |-  ( ph  ->  ( F  e.  dom  ( ~~> u `  S )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  (
 ( ( F `  k ) `  z
 )  -  ( ( F `  j ) `
  z ) ) )  <  x ) )
 
Theoremulmcau2 19773* A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 
0  <  x there is a  j such that for all  j  <_  k ,  m the functions  F ( k ) and  F ( m ) are uniformly within  x of each other on  S. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   =>    |-  ( ph  ->  ( F  e.  dom  ( ~~> u `  S )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. m  e.  ( ZZ>= `  k ) A. z  e.  S  ( abs `  ( (
 ( F `  k
 ) `  z )  -  ( ( F `  m ) `  z
 ) ) )  < 
 x ) )
 
Theoremulmss 19774* A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  T 
 C_  S )   &    |-  (
 ( ph  /\  x  e.  Z )  ->  A  e.  W )   &    |-  ( ph  ->  ( x  e.  Z  |->  A ) ( ~~> u `  S ) G )   =>    |-  ( ph  ->  ( x  e.  Z  |->  ( A  |`  T ) ) ( ~~> u `  T ) ( G  |`  T ) )
 
Theoremulmbdd 19775* A uniform limit of bounded functions is bounded. (Contributed by Mario Carneiro, 27-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  E. x  e.  RR  A. z  e.  S  ( abs `  ( ( F `  k ) `  z ) )  <_  x )   &    |-  ( ph  ->  F ( ~~> u `  S ) G )   =>    |-  ( ph  ->  E. x  e.  RR  A. z  e.  S  ( abs `  ( G `  z ) ) 
 <_  x )
 
Theoremulmcn 19776 A uniform limit of continuous functions is continuous. (Contributed by Mario Carneiro, 27-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( S -cn-> CC ) )   &    |-  ( ph  ->  F ( ~~> u `  S ) G )   =>    |-  ( ph  ->  G  e.  ( S -cn-> CC ) )
 
Theoremulmdvlem1 19777* Lemma for ulmdv 19780. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  X ) )   &    |-  ( ph  ->  G : X --> CC )   &    |-  (
 ( ph  /\  z  e.  X )  ->  (
 k  e.  Z  |->  ( ( F `  k
 ) `  z )
 )  ~~>  ( G `  z ) )   &    |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) ) ( ~~> u `  X ) H )   &    |-  ( ( ph  /\  ps )  ->  C  e.  X )   &    |-  ( ( ph  /\  ps )  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ps )  ->  U  e.  RR+ )   &    |-  ( ( ph  /\  ps )  ->  W  e.  RR+ )   &    |-  ( ( ph  /\  ps )  ->  U  <  W )   &    |-  ( ( ph  /\  ps )  ->  ( C (
 ball `  ( ( abs 
 o.  -  )  |`  ( S  X.  S ) ) ) U )  C_  X )   &    |-  ( ( ph  /\ 
 ps )  ->  ( abs `  ( Y  -  C ) )  <  U )   &    |-  ( ( ph  /\ 
 ps )  ->  N  e.  Z )   &    |-  ( ( ph  /\ 
 ps )  ->  A. m  e.  ( ZZ>= `  N ) A. x  e.  X  ( abs `  ( (
 ( S  _D  ( F `  N ) ) `
  x )  -  ( ( S  _D  ( F `  m ) ) `  x ) ) )  <  (
 ( R  /  2
 )  /  2 )
 )   &    |-  ( ( ph  /\  ps )  ->  ( abs `  (
 ( ( S  _D  ( F `  N ) ) `  C )  -  ( H `  C ) ) )  <  ( R  / 
 2 ) )   &    |-  (
 ( ph  /\  ps )  ->  Y  e.  X )   &    |-  ( ( ph  /\  ps )  ->  Y  =/=  C )   &    |-  ( ( ph  /\  ps )  ->  ( ( abs `  ( Y  -  C ) )  <  W  ->  ( abs `  ( (
 ( ( ( F `
  N ) `  Y )  -  (
 ( F `  N ) `  C ) ) 
 /  ( Y  -  C ) )  -  ( ( S  _D  ( F `  N ) ) `  C ) ) )  <  (
 ( R  /  2
 )  /  2 )
 ) )   =>    |-  ( ( ph  /\  ps )  ->  ( abs `  (
 ( ( ( G `
  Y )  -  ( G `  C ) )  /  ( Y  -  C ) )  -  ( H `  C ) ) )  <  R )
 
Theoremulmdvlem2 19778* Lemma for ulmdv 19780. (Contributed by Mario Carneiro, 8-May-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  X ) )   &    |-  ( ph  ->  G : X --> CC )   &    |-  (
 ( ph  /\  z  e.  X )  ->  (
 k  e.  Z  |->  ( ( F `  k
 ) `  z )
 )  ~~>  ( G `  z ) )   &    |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) ) ( ~~> u `  X ) H )   =>    |-  ( ( ph  /\  k  e.  Z )  ->  dom  ( S  _D  ( F `  k ) )  =  X )
 
Theoremulmdvlem3 19779* Lemma for ulmdv 19780. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  X ) )   &    |-  ( ph  ->  G : X --> CC )   &    |-  (
 ( ph  /\  z  e.  X )  ->  (
 k  e.  Z  |->  ( ( F `  k
 ) `  z )
 )  ~~>  ( G `  z ) )   &    |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) ) ( ~~> u `  X ) H )   =>    |-  ( ( ph  /\  z  e.  X )  ->  z
 ( S  _D  G ) ( H `  z ) )
 
Theoremulmdv 19780* If  F is a sequence of differentiable functions on  X which converge pointwise to  G, and the derivatives of 
F ( n ) converge uniformly to  H, then  G is differentiable with derivative  H. (Contributed by Mario Carneiro, 27-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  X ) )   &    |-  ( ph  ->  G : X --> CC )   &    |-  (
 ( ph  /\  z  e.  X )  ->  (
 k  e.  Z  |->  ( ( F `  k
 ) `  z )
 )  ~~>  ( G `  z ) )   &    |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) ) ( ~~> u `  X ) H )   =>    |-  ( ph  ->  ( S  _D  G )  =  H )
 
Theoremmtest 19781* The Weierstrass M-test. If  F is a sequence of functions which are uniformly bounded by the convergent sequence  M ( k ), then the series generated by the sequence  F converges uniformly. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  Z  =  ( ZZ>= `  N )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : Z --> ( CC  ^m  S ) )   &    |-  ( ph  ->  M  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( M `  k )  e. 
 RR )   &    |-  ( ( ph  /\  ( k  e.  Z  /\  z  e.  S ) )  ->  ( abs `  ( ( F `  k ) `  z
 ) )  <_  ( M `  k ) )   &    |-  ( ph  ->  seq  N (  +  ,  M )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  N (  o F  +  ,  F )  e.  dom  (
 ~~> u `  S ) )
 
Theoremmbfulm 19782 A uniform limit of measurable functions is measurable. (This is just a corollary of the fact that a pointwise limit of measurable functions is measurable, see mbflim 19023.) (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z -->MblFn )   &    |-  ( ph  ->  F ( ~~> u `  S ) G )   =>    |-  ( ph  ->  G  e. MblFn )
 
Theoremiblulm 19783 A uniform limit of integrable functions is integrable. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> L ^1 )   &    |-  ( ph  ->  F ( ~~> u `  S ) G )   &    |-  ( ph  ->  ( vol `  S )  e.  RR )   =>    |-  ( ph  ->  G  e.  L ^1 )
 
Theoremitgulm 19784* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> L ^1 )   &    |-  ( ph  ->  F ( ~~> u `  S ) G )   &    |-  ( ph  ->  ( vol `  S )  e.  RR )   =>    |-  ( ph  ->  (
 k  e.  Z  |->  S. S ( ( F `
  k ) `  x )  _d x ) 
 ~~>  S. S ( G `
  x )  _d x )
 
Theoremitgulm2 19785* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( x  e.  S  |->  A )  e.  ( S -cn-> CC ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( x  e.  S  |->  A )  e.  L ^1 )   &    |-  ( ph  ->  ( k  e.  Z  |->  ( x  e.  S  |->  A ) ) ( ~~> u `  S ) ( x  e.  S  |->  B ) )   &    |-  ( ph  ->  ( vol `  S )  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  S  |->  B )  e.  L ^1  /\  ( k  e.  Z  |->  S. S A  _d x )  ~~>  S. S B  _d x ) )
 
13.2.3  Power series
 
Theorempserval 19786* Value of the function  G that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   =>    |-  ( X  e.  CC  ->  ( G `  X )  =  ( m  e.  NN0  |->  ( ( A `
  m )  x.  ( X ^ m ) ) ) )
 
Theorempserval2 19787* Value of the function  G that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   =>    |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  ( ( G `
  X ) `  N )  =  (
 ( A `  N )  x.  ( X ^ N ) ) )
 
Theorempsergf 19788* The sequence of terms in the infinite sequence defining a power series for fixed  X. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   =>    |-  ( ph  ->  ( G `  X ) :
 NN0
 --> CC )
 
Theoremradcnvlem1 19789* Lemma for radcnvlt1 19794, radcnvle 19796. If  X is a point closer to zero than  Y and the power series converges at 
Y, then it converges absolutely at 
X, even if the terms in the sequence are multiplied by  n. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  ( abs `  Y ) )   &    |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e.  dom  ~~>  )   &    |-  H  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
  X ) `  m ) ) ) )   =>    |-  ( ph  ->  seq  0
 (  +  ,  H )  e.  dom  ~~>  )
 
Theoremradcnvlem2 19790* Lemma for radcnvlt1 19794, radcnvle 19796. If  X is a point closer to zero than  Y and the power series converges at 
Y, then it converges absolutely at  X. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  ( abs `  Y ) )   &    |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  0 (  +  ,  ( abs  o.  ( G `
  X ) ) )  e.  dom  ~~>  )
 
Theoremradcnvlem3 19791* Lemma for radcnvlt1 19794, radcnvle 19796. If  X is a point closer to zero than  Y and the power series converges at 
Y, then it converges at  X. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  ( abs `  Y ) )   &    |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  0 (  +  ,  ( G `  X ) )  e.  dom  ~~>  )
 
Theoremradcnv0 19792* Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   =>    |-  ( ph  ->  0  e.  {
 r  e.  RR  |  seq  0 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } )
 
Theoremradcnvcl 19793* The radius of convergence  R of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   =>    |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
 
Theoremradcnvlt1 19794* If  X is within the open disk of radius  R centered at zero, then the infinite series converges absolutely at  X, and also converges when the series is multiplied by  n. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  R )   &    |-  H  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  (
 ( G `  X ) `  m ) ) ) )   =>    |-  ( ph  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\ 
 seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  ) )
 
Theoremradcnvlt2 19795* If  X is within the open disk of radius  R centered at zero, then the infinite series converges at  X. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  R )   =>    |-  ( ph  ->  seq  0
 (  +  ,  ( G `  X ) )  e.  dom  ~~>  )
 
Theoremradcnvle 19796* If  X is a convergent point of the infinite series, then 
X is within the closed disk of radius  R centered at zero. Or, by contraposition, the series divergers at any point strictly more than  R from the origin. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  seq  0 (  +  ,  ( G `  X ) )  e.  dom  ~~>  )   =>    |-  ( ph  ->  ( abs `  X )  <_  R )
 
Theoremdvradcnv 19797* The radius of convergence of the (formal) derivative  H of the power series  G is at least as large as the radius of convergence of  G. (In fact they are equal, but we don't have as much use for the negative side of this claim.) (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  H  =  ( n  e.  NN0  |->  ( ( ( n  +  1 )  x.  ( A `  ( n  +  1 )
 ) )  x.  ( X ^ n ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  R )   =>    |-  ( ph  ->  seq  0
 (  +  ,  H )  e.  dom  ~~>  )
 
Theorempserulm 19798* If  S is a region contained in a circle of radius  M  <  R, then the sequence of partial sums of the infinite series converges uniformly on  S. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  H  =  ( i  e.  NN0  |->  ( y  e.  S  |->  (  seq  0 (  +  ,  ( G `  y
 ) ) `  i
 ) ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  M  <  R )   &    |-  ( ph  ->  S 
 C_  ( `' abs " ( 0 [,] M ) ) )   =>    |-  ( ph  ->  H ( ~~> u `  S ) F )
 
Theorempsercn2 19799* Since by pserulm 19798 the series converges uniformly, it is also continuous by ulmcn 19776. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  H  =  ( i  e.  NN0  |->  ( y  e.  S  |->  (  seq  0 (  +  ,  ( G `  y
 ) ) `  i
 ) ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  M  <  R )   &    |-  ( ph  ->  S 
 C_  ( `' abs " ( 0 [,] M ) ) )   =>    |-  ( ph  ->  F  e.  ( S -cn-> CC ) )
 
Theorempsercnlem2 19800* Lemma for psercn 19802. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  S  =  ( `' abs " (
 0 [,) R ) )   &    |-  ( ( ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  <  M  /\  M  <  R ) )   =>    |-  ( ( ph  /\  a  e.  S )  ->  (
 a  e.  ( 0 ( ball `  ( abs  o. 
 -  ) ) M )  /\  ( 0 ( ball `  ( abs  o. 
 -  ) ) M )  C_  ( `' abs " ( 0 [,]
 M ) )  /\  ( `' abs " ( 0 [,] M ) ) 
 C_  S ) )
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