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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremulmcl 23201 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( F ( ~~> u `  S ) G  ->  G : S --> CC )
 
Theoremulmf 23202* 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 23203 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 23204 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 23205* Simplify ulmval 23200 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 23206* 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 23207* 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 23208 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 23209* Lemma for ulmshft 23210. (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 23210* 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 23211 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 )
 
Theoremulmuni 23212 An sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.)
 |-  ( ( F ( ~~> u `  S ) G  /\  F ( ~~> u `  S ) H )  ->  G  =  H )
 
Theoremulmdm 23213 Two ways to express that a function has a limit. (The expression  ( ( ~~> u `  S ) `  F ) is sometimes useful as a shorthand for "the unique limit of the function 
F"). (Contributed by Mario Carneiro, 5-Jul-2017.)
 |-  ( F  e.  dom  (
 ~~> u `  S )  <->  F ( ~~> u `  S ) ( ( ~~> u `  S ) `
  F ) )
 
Theoremulmcaulem 23214* Lemma for ulmcau 23215 and ulmcau2 23216: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 13397. (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 23215* 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 23216 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 23216* 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 23217* 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 23218* 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 23219 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 23220* Lemma for ulmdv 23223. (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 23221* Lemma for ulmdv 23223. (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 23222* Lemma for ulmdv 23223. (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 23223* 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 23224* 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 (  oF  +  ,  F )  e.  dom  (
 ~~> u `  S ) )
 
Theoremmtestbdd 23225* Given the hypotheses of the Weierstrass M-test, the convergent function of the sequence is uniformly bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  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 (  oF  +  ,  F )
 ( ~~> u `  S ) T )   =>    |-  ( ph  ->  E. x  e.  RR  A. z  e.  S  ( abs `  ( T `  z ) ) 
 <_  x )
 
Theoremmbfulm 23226 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 22503.) (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 23227 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 23228* 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 23229* 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 ) )
 
14.2.3  Power series
 
Theorempserval 23230* 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 23231* 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 23232* 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 23233* Lemma for radcnvlt1 23238, radcnvle 23240. 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 23234* Lemma for radcnvlt1 23238, radcnvle 23240. 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 23235* Lemma for radcnvlt1 23238, radcnvle 23240. 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 23236* 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 23237* 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 23238* 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 23239* 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 23240* 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 diverges 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 23241* 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 23242* 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 23243* Since by pserulm 23242 the series converges uniformly, it is also continuous by ulmcn 23219. (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 23244* Lemma for psercn 23246. (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 ) )
 
Theorempsercnlem1 23245* Lemma for psercn 23246. (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 ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   =>    |-  ( ( ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  <  M  /\  M  <  R ) )
 
Theorempsercn 23246* An infinite series converges to a continuous function on the open disk of radius  R, where  R is the radius of convergence of the series. (Contributed by Mario Carneiro, 4-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 ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   =>    |-  ( ph  ->  F  e.  ( S -cn-> CC )
 )
 
Theorempserdvlem1 23247* Lemma for pserdv 23249. (Contributed by Mario Carneiro, 7-May-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 ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   =>    |-  ( ( ph  /\  a  e.  S )  ->  (
 ( ( ( abs `  a )  +  M )  /  2 )  e.  RR+  /\  ( abs `  a
 )  <  ( (
 ( abs `  a )  +  M )  /  2
 )  /\  ( (
 ( abs `  a )  +  M )  /  2
 )  <  R )
 )
 
Theorempserdvlem2 23248* Lemma for pserdv 23249. (Contributed by Mario Carneiro, 7-May-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 ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   &    |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M )  /  2 ) )   =>    |-  ( ( ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  =  ( y  e.  B  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1
 ) ) )  x.  ( y ^ k
 ) ) ) )
 
Theorempserdv 23249* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-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 ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   &    |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M )  /  2 ) )   =>    |-  ( ph  ->  ( CC  _D  F )  =  ( y  e.  S  |->  sum_ k  e.  NN0  ( (
 ( k  +  1 )  x.  ( A `
  ( k  +  1 ) ) )  x.  ( y ^
 k ) ) ) )
 
Theorempserdv2 23250* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-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 ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   &    |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M )  /  2 ) )   =>    |-  ( ph  ->  ( CC  _D  F )  =  ( y  e.  S  |->  sum_ k  e.  NN  ( ( k  x.  ( A `
  k ) )  x.  ( y ^
 ( k  -  1
 ) ) ) ) )
 
Theoremabelthlem1 23251* Lemma for abelth 23261. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   =>    |-  ( ph  ->  1  <_  sup ( { r  e. 
 RR  |  seq 0
 (  +  ,  (
 ( z  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( z ^ n ) ) ) ) `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )
 
Theoremabelthlem2 23252* Lemma for abelth 23261. The peculiar region  S, known as a Stolz angle , is a teardrop-shaped subset of the closed unit ball containing  1. Indeed, except for  1 itself, the rest of the Stolz angle is enclosed in the open unit ball. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   =>    |-  ( ph  ->  (
 1  e.  S  /\  ( S  \  { 1 } )  C_  (
 0 ( ball `  ( abs  o.  -  ) ) 1 ) ) )
 
Theoremabelthlem3 23253* Lemma for abelth 23261. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   =>    |-  ( ( ph  /\  X  e.  S )  ->  seq 0
 (  +  ,  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( X ^ n ) ) ) )  e.  dom  ~~>  )
 
Theoremabelthlem4 23254* Lemma for abelth 23261. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   =>    |-  ( ph  ->  F : S --> CC )
 
Theoremabelthlem5 23255* Lemma for abelth 23261. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  ~~>  0 )   =>    |-  (
 ( ph  /\  X  e.  ( 0 ( ball `  ( abs  o.  -  ) ) 1 ) )  ->  seq 0
 (  +  ,  (
 k  e.  NN0  |->  ( ( 
 seq 0 (  +  ,  A ) `  k
 )  x.  ( X ^ k ) ) ) )  e.  dom  ~~>  )
 
Theoremabelthlem6 23256* Lemma for abelth 23261. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  ~~>  0 )   &    |-  ( ph  ->  X  e.  ( S  \  { 1 } ) )   =>    |-  ( ph  ->  ( F `  X )  =  ( ( 1  -  X )  x.  sum_ n  e.  NN0  ( (  seq 0 (  +  ,  A ) `  n )  x.  ( X ^ n ) ) ) )
 
Theoremabelthlem7a 23257* Lemma for abelth 23261. (Contributed by Mario Carneiro, 8-May-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  ~~>  0 )   &    |-  ( ph  ->  X  e.  ( S  \  { 1 } ) )   =>    |-  ( ph  ->  ( X  e.  CC  /\  ( abs `  ( 1  -  X ) )  <_  ( M  x.  (
 1  -  ( abs `  X ) ) ) ) )
 
Theoremabelthlem7 23258* Lemma for abelth 23261. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  ~~>  0 )   &    |-  ( ph  ->  X  e.  ( S  \  { 1 } ) )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A. k  e.  ( ZZ>= `  N ) ( abs `  (  seq 0
 (  +  ,  A ) `  k ) )  <  R )   &    |-  ( ph  ->  ( abs `  (
 1  -  X ) )  <  ( R 
 /  ( sum_ n  e.  ( 0 ... ( N  -  1 ) ) ( abs `  (  seq 0 (  +  ,  A ) `  n ) )  +  1
 ) ) )   =>    |-  ( ph  ->  ( abs `  ( F `  X ) )  < 
 ( ( M  +  1 )  x.  R ) )
 
Theoremabelthlem8 23259* Lemma for abelth 23261. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  ~~>  0 )   =>    |-  (
 ( ph  /\  R  e.  RR+ )  ->  E. w  e.  RR+  A. y  e.  S  ( ( abs `  (
 1  -  y ) )  <  w  ->  ( abs `  ( ( F `  1 )  -  ( F `  y ) ) )  <  R ) )
 
Theoremabelthlem9 23260* Lemma for abelth 23261. By adjusting the constant term, we can assume that the entire series converges to 
0. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   =>    |-  ( ( ph  /\  R  e.  RR+ )  ->  E. w  e.  RR+  A. y  e.  S  ( ( abs `  (
 1  -  y ) )  <  w  ->  ( abs `  ( ( F `  1 )  -  ( F `  y ) ) )  <  R ) )
 
Theoremabelth 23261* Abel's theorem. If the power series  sum_ n  e.  NN0 A
( n ) ( x ^ n ) is convergent at  1, then it is equal to the limit from "below", along a Stolz angle  S (note that the  M  =  1 case of a Stolz angle is the real line  [ 0 ,  1 ]). (Continuity on  S  \  { 1 } follows more generally from psercn 23246.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0 
 <_  M )   &    |-  S  =  {
 z  e.  CC  |  ( abs `  ( 1  -  z ) )  <_  ( M  x.  (
 1  -  ( abs `  z ) ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   =>    |-  ( ph  ->  F  e.  ( S -cn-> CC ) )
 
Theoremabelth2 23262* Abel's theorem, restricted to the 
[ 0 ,  1 ] interval. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq 0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  F  =  ( x  e.  ( 0 [,] 1
 )  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   =>    |-  ( ph  ->  F  e.  ( ( 0 [,] 1 ) -cn-> CC )
 )
 
14.3  Basic trigonometry
 
14.3.1  The exponential, sine, and cosine functions (cont.)
 
Theoremefcn 23263 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |- 
 exp  e.  ( CC -cn-> CC )
 
Theoremsincn 23264 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
 |- 
 sin  e.  ( CC -cn-> CC )
 
Theoremcoscn 23265 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
 |- 
 cos  e.  ( CC -cn-> CC )
 
Theoremreeff1olem 23266* Lemma for reeff1o 23267. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( ( U  e.  RR  /\  1  <  U )  ->  E. x  e.  RR  ( exp `  x )  =  U )
 
Theoremreeff1o 23267 The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( exp  |`  RR ) : RR
 -1-1-onto-> RR+
 
Theoremreefiso 23268 The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.)
 |-  ( exp  |`  RR )  Isom  <  ,  <  ( RR ,  RR+ )
 
Theoremefcvx 23269 The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
 ) )  ->  ( exp `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) ) )  < 
 ( ( T  x.  ( exp `  A )
 )  +  ( ( 1  -  T )  x.  ( exp `  B ) ) ) )
 
Theoremreefgim 23270 The exponential function is a group isomorphism from the group of reals under addition to the group of positive reals under multiplication. (Contributed by Mario Carneiro, 21-Jun-2015.) (Revised by Thierry Arnoux, 30-Jun-2019.)
 |-  P  =  ( (mulGrp ` fld )s  RR+ )   =>    |-  ( exp  |`  RR )  e.  (RRfld GrpIso  P )
 
14.3.2  Properties of pi = 3.14159...
 
Theorempilem1 23271 Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( A  e.  ( RR+ 
 i^i  ( `' sin " { 0 } )
 ) 
 <->  ( A  e.  RR+  /\  ( sin `  A )  =  0 )
 )
 
Theorempilem2 23272 Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.)
 |-  ( ph  ->  A  e.  ( 2 (,) 4
 ) )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( sin `  A )  =  0 )   &    |-  ( ph  ->  ( sin `  B )  =  0 )   &    |-  ( ph  ->  pi  <  A )   =>    |-  ( ph  ->  (
 ( pi  +  A )  /  2 )  <_  B )
 
Theorempilem2OLD 23273 Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. (Contributed by Mario Carneiro, 12-Jun-2014.) Obsolete version of pilem2 23272 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A  e.  ( 2 (,) 4
 ) )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( sin `  A )  =  0 )   &    |-  ( ph  ->  ( sin `  B )  =  0 )   &    |-  ( ph  ->  pi  <  A )   =>    |-  ( ph  ->  (
 ( pi  +  A )  /  2 )  <_  B )
 
Theorempilem3 23274 Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) (Revised by AV, 14-Sep-2020.)
 |-  ( pi  e.  (
 2 (,) 4 )  /\  ( sin `  pi )  =  0 )
 
Theorempilem3OLD 23275 Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) Obsolete version of pilem3 23274 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( pi  e.  (
 2 (,) 4 )  /\  ( sin `  pi )  =  0 )
 
Theorempigt2lt4 23276  pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  ( 2  <  pi  /\  pi  <  4 )
 
Theoremsinpi 23277 The sine of  pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  pi )  =  0
 
Theorempire 23278  pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  pi  e.  RR
 
Theorempicn 23279  pi is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  pi  e.  CC
 
Theorempipos 23280  pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  0  <  pi
 
Theorempirp 23281  pi is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  pi  e.  RR+
 
Theoremnegpicn 23282  -u pi is a real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u pi  e.  CC
 
Theoremsinhalfpilem 23283 Lemma for sinhalfpi 23288 and coshalfpi 23289. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( ( sin `  ( pi  /  2 ) )  =  1  /\  ( cos `  ( pi  / 
 2 ) )  =  0 )
 
Theoremhalfpire 23284  pi  /  2 is real. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( pi  /  2
 )  e.  RR
 
Theoremneghalfpire 23285  -u pi  / 
2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u ( pi  /  2
 )  e.  RR
 
Theoremneghalfpirx 23286  -u pi  / 
2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u ( pi  /  2
 )  e.  RR*
 
Theorempidiv2halves 23287 Adding  pi  /  2 to itself is  pi (common case). See 2halves 10841. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ( pi  / 
 2 )  +  ( pi  /  2 ) )  =  pi
 
Theoremsinhalfpi 23288 The sine of  pi  /  2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  ( pi  /  2 ) )  =  1
 
Theoremcoshalfpi 23289 The cosine of  pi  /  2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  ( pi  /  2 ) )  =  0
 
Theoremcosneghalfpi 23290 The cosine of  -u pi  /  2 is zero. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( cos `  -u ( pi  /  2 ) )  =  0
 
Theoremefhalfpi 23291 The exponential of  _i pi  /  2 is  _i. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( exp `  ( _i  x.  ( pi  / 
 2 ) ) )  =  _i
 
Theoremcospi 23292 The cosine of  pi is  -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  pi )  =  -u 1
 
Theoremefipi 23293 The exponential of  _i  x.  pi. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( exp `  ( _i  x.  pi ) )  =  -u 1
 
Theoremeulerid 23294 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  ( ( exp `  ( _i  x.  pi ) )  +  1 )  =  0
 
Theoremsin2pi 23295 The sine of  2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  (
 2  x.  pi ) )  =  0
 
Theoremcos2pi 23296 The cosine of  2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  (
 2  x.  pi ) )  =  1
 
Theoremef2pi 23297 The exponential of  2 pi _i is  1. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( exp `  ( _i  x.  ( 2  x.  pi ) ) )  =  1
 
Theoremef2kpi 23298 The exponential of  2 K pi _i is  1. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( K  e.  ZZ  ->  ( exp `  (
 ( _i  x.  (
 2  x.  pi ) )  x.  K ) )  =  1 )
 
Theoremefper 23299 The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  ( A  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) ) )  =  ( exp `  A ) )
 
Theoremsinperlem 23300 Lemma for sinper 23301 and cosper 23302. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( F `  A )  =  ( (
 ( exp `  ( _i  x.  A ) ) O ( exp `  ( -u _i  x.  A ) ) )  /  D ) )   &    |-  ( ( A  +  ( K  x.  ( 2  x.  pi ) ) )  e. 
 CC  ->  ( F `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( ( ( exp `  ( _i  x.  ( A  +  ( K  x.  (
 2  x.  pi ) ) ) ) ) O ( exp `  ( -u _i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) ) )  /  D ) )   =>    |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( F `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( F `
  A ) )
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