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Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqshash 12301* The cardinality of a set with an equivalence relation is the sum of the cardinalities of its equivalence classes. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( ph  ->  .~  Er  A )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  ( # `  A )  = 
 sum_ x  e.  ( A /.  .~  ) ( # `  x ) )
 
Theoremackbijnn 12302* Translate the Ackermann bijection ackbij1 7880 onto the natural numbers. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  F  =  ( x  e.  ( ~P NN0  i^i 
 Fin )  |->  sum_ y  e.  x  ( 2 ^ y ) )   =>    |-  F : ( ~P NN0  i^i 
 Fin ) -1-1-onto-> NN0
 
5.8.4  The binomial theorem
 
Theorembinomlem 12303* Lemma for binom 12304 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ps  ->  (
 ( A  +  B ) ^ N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A ^ ( N  -  k ) )  x.  ( B ^
 k ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( A  +  B ) ^
 ( N  +  1 ) )  =  sum_ k  e.  ( 0 ... ( N  +  1 ) ) ( ( ( N  +  1 )  _C  k )  x.  ( ( A ^ ( ( N  +  1 )  -  k ) )  x.  ( B ^ k
 ) ) ) )
 
Theorembinom 12304* The binomial theorem:  ( A  +  B
) ^ N is the sum from  k  =  0 to  N of  ( N  _C  k )  x.  ( ( A ^
k )  x.  ( B ^ ( N  -  k ) ). Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 12303. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) ^ N )  =  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  ( ( A ^
 ( N  -  k
 ) )  x.  ( B ^ k ) ) ) )
 
Theorembinom1p 12305* Special case of the binomial theorem for  ( 1  +  A
) ^ N. (Contributed by Paul Chapman, 10-May-2007.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( ( 1  +  A ) ^ N )  =  sum_ k  e.  ( 0 ...
 N ) ( ( N  _C  k )  x.  ( A ^
 k ) ) )
 
Theorembinom11 12306* Special case of the binomial theorem for  2 ^ N. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( N  e.  NN0  ->  ( 2 ^ N )  =  sum_ k  e.  ( 0 ... N ) ( N  _C  k ) )
 
Theorembinom1dif 12307* A summation for the difference between  ( ( A  + 
1 ) ^ N
) and  ( A ^ N ). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( ( ( A  +  1 ) ^ N )  -  ( A ^ N ) )  =  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k ) ) )
 
Theorembcxmaslem1 12308 Lemma for bcxmas 12310. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( A  =  B  ->  ( ( N  +  A )  _C  A )  =  ( ( N  +  B )  _C  B ) )
 
Theorembcxmaslem2 12309 Lemma for bcxmas 12310. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C )
 )  =  ( ( A  +  C )  +  B ) )
 
Theorembcxmas 12310* Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( ( N  +  1 )  +  M )  _C  M )  =  sum_ j  e.  (
 0 ... M ) ( ( N  +  j
 )  _C  j )
 )
 
5.8.5  The inclusion/exclusion principle
 
Theoremincexclem 12311* Lemma for incexc 12312. (Contributed by Mario Carneiro, 7-Aug-2017.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  B )  -  ( # `  ( B  i^i  U. A ) ) )  =  sum_ s  e.  ~P  A ( (
 -u 1 ^ ( # `
  s ) )  x.  ( # `  ( B  i^i  |^| s ) ) ) )
 
Theoremincexc 12312* The inclusion/exclusion principle for counting the elements of a finite union of finite sets. (Contributed by Mario Carneiro, 7-Aug-2017.)
 |-  ( ( A  e.  Fin  /\  A  C_  Fin )  ->  ( # `  U. A )  =  sum_ s  e.  ( ~P A  \  { (/) } ) ( ( -u 1 ^ (
 ( # `  s )  -  1 ) )  x.  ( # `  |^| s
 ) ) )
 
Theoremincexc2 12313* The inclusion/exclusion principle for counting the elements of a finite union of finite sets. (Contributed by Mario Carneiro, 7-Aug-2017.)
 |-  ( ( A  e.  Fin  /\  A  C_  Fin )  ->  ( # `  U. A )  =  sum_ n  e.  ( 1 ... ( # `
  A ) ) ( ( -u 1 ^ ( n  -  1 ) )  x. 
 sum_ s  e.  { k  e.  ~P A  |  ( # `  k )  =  n }  ( # ` 
 |^| s ) ) )
 
5.8.6  Infinite sums (cont.)
 
Theoremisumshft 12314* Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  ( M  +  K ) )   &    |-  (
 j  =  ( K  +  k )  ->  A  =  B )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  W ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  W  A  =  sum_ k  e.  Z  B )
 
Theoremisumsplit 12315* Split off the first  N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  =  (
 sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A ) )
 
Theoremisum1p 12316* The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  =  ( ( F `  M )  +  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) A ) )
 
Theoremisumnn0nn 12317* Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( k  =  0 
 ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  A  e.  CC )   &    |-  ( ph  ->  seq  0 (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  NN0  A  =  ( B  +  sum_ k  e.  NN  A ) )
 
Theoremisumrpcl 12318* The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR+ )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  W  A  e.  RR+ )
 
Theoremisumle 12319* Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  <_  B )   &    |-  ( ph  ->  seq 
 M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  M (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  <_  sum_ k  e.  Z  B )
 
Theoremisumless 12320* A finite sum of nonnegative numbers is less or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  B )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <_  sum_ k  e.  Z  B )
 
Theoremisumsup2 12321* An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  seq  M (  +  ,  F )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  A )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  A )   &    |-  ( ph  ->  E. x  e.  RR  A. j  e.  Z  ( G `  j )  <_  x )   =>    |-  ( ph  ->  G  ~~>  sup ( ran 
 G ,  RR ,  <  ) )
 
Theoremisumsup 12322* An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  seq  M (  +  ,  F )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  A )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  A )   &    |-  ( ph  ->  E. x  e.  RR  A. j  e.  Z  ( G `  j )  <_  x )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  =  sup ( ran  G ,  RR ,  <  ) )
 
Theoremisumltss 12323* A partial sum of a series with positive terms is less than the infinite sum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  B  e.  RR+ )   &    |-  ( ph  ->  seq 
 M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B )
 
Theoremclimcndslem1 12324* Lemma for climcnds 12326: bound the original series by the condensed series. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  NN )  ->  0  <_  ( F `  k ) )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  n  e.  NN0 )  ->  ( G `  n )  =  ( (
 2 ^ n )  x.  ( F `  ( 2 ^ n ) ) ) )   =>    |-  ( ( ph  /\  N  e.  NN0 )  ->  (  seq  1 (  +  ,  F ) `  (
 ( 2 ^ ( N  +  1 )
 )  -  1 ) )  <_  (  seq  0 (  +  ,  G ) `  N ) )
 
Theoremclimcndslem2 12325* Lemma for climcnds 12326: bound the condensed series by the original series. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  NN )  ->  0  <_  ( F `  k ) )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  n  e.  NN0 )  ->  ( G `  n )  =  ( (
 2 ^ n )  x.  ( F `  ( 2 ^ n ) ) ) )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  (  seq  1 (  +  ,  G ) `  N )  <_  ( 2  x.  (  seq  1 (  +  ,  F ) `
  ( 2 ^ N ) ) ) )
 
Theoremclimcnds 12326* The Cauchy condensation test. If  a ( k ) is a decreasing sequence of nonnegative terms, then  sum_ k  e.  NN a ( k ) converges iff  sum_ n  e. 
NN0 2 ^ n  x.  a ( 2 ^ n ) converges. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  NN )  ->  0  <_  ( F `  k ) )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  n  e.  NN0 )  ->  ( G `  n )  =  ( (
 2 ^ n )  x.  ( F `  ( 2 ^ n ) ) ) )   =>    |-  ( ph  ->  (  seq  1 (  +  ,  F )  e.  dom  ~~>  <->  seq  0 (  +  ,  G )  e.  dom  ~~>  ) )
 
5.8.7  Miscellaneous converging and diverging sequences
 
Theoremdivrcnv 12327* The sequence of reciprocals of real numbers, multiplied by the factor  A, converges to zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  CC  ->  ( n  e.  RR+  |->  ( A  /  n ) )  ~~> r  0 )
 
Theoremdivcnv 12328* The sequence of reciprocals of natural numbers, multiplied by the factor  A, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  CC  ->  ( n  e.  NN  |->  ( A  /  n ) )  ~~>  0 )
 
Theoremflo1 12329 The floor function satisfies  |_ ( x )  =  x  +  O
( 1 ). (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( x  e.  RR  |->  ( x  -  ( |_ `  x ) ) )  e.  O ( 1 )
 
Theoremsupcvg 12330* Extract a sequence  f in  X such that the image of the points in the bounded set  A converges to the supremum  S of the set. Similar to Equation 4 of [Kreyszig] p. 144. The proof uses countable choice ax-cc 8077. (Contributed by Mario Carneiro, 15-Feb-2013.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
 |-  X  e.  _V   &    |-  S  =  sup ( A ,  RR ,  <  )   &    |-  R  =  ( n  e.  NN  |->  ( S  -  (
 1  /  n )
 ) )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  ->  F : X -onto-> A )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |-  ( ph  ->  E. f
 ( f : NN --> X  /\  ( F  o.  f )  ~~>  S ) )
 
Theoreminfcvgaux1i 12331* Auxiliary theorem for applications of supcvg 12330. Hypothesis for several supremum theorems. (Contributed by NM, 8-Feb-2008.)
 |-  R  =  { x  |  E. y  e.  X  x  =  -u A }   &    |-  (
 y  e.  X  ->  A  e.  RR )   &    |-  Z  e.  X   &    |-  E. z  e. 
 RR  A. w  e.  R  w  <_  z   =>    |-  ( R  C_  RR  /\  R  =/=  (/)  /\  E. z  e.  RR  A. w  e.  R  w  <_  z
 )
 
Theoreminfcvgaux2i 12332* Auxiliary theorem for applications of supcvg 12330. (Contributed by NM, 4-Mar-2008.)
 |-  R  =  { x  |  E. y  e.  X  x  =  -u A }   &    |-  (
 y  e.  X  ->  A  e.  RR )   &    |-  Z  e.  X   &    |-  E. z  e. 
 RR  A. w  e.  R  w  <_  z   &    |-  S  =  -u sup ( R ,  RR ,  <  )   &    |-  ( y  =  C  ->  A  =  B )   =>    |-  ( C  e.  X  ->  S  <_  B )
 
Theoremharmonic 12333 The harmonic series  H diverges. This fact follows from the stronger emcl 20312, which establishes that the harmonic series grows as  log n  +  gamma  + o(1), but this uses a more elementary method, attributed to Nicole Oresme (1323-1382). (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( 1 
 /  n ) )   &    |-  H  =  seq  1 (  +  ,  F )   =>    |-  -.  H  e.  dom  ~~>
 
5.8.8  Arithmetic series
 
Theoremarisum 12334* Arithmetic series sum of the first 
N positive integers. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 1
 ... N ) k  =  ( ( ( N ^ 2 )  +  N )  / 
 2 ) )
 
Theoremarisum2 12335* Arithmetic series sum of the first 
N nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0
 ... ( N  -  1 ) ) k  =  ( ( ( N ^ 2 )  -  N )  / 
 2 ) )
 
Theoremtrireciplem 12336 Lemma for trirecip 12337. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  F  =  ( n  e.  NN  |->  ( 1 
 /  ( n  x.  ( n  +  1
 ) ) ) )   =>    |-  seq  1 (  +  ,  F )  ~~>  1
 
Theoremtrirecip 12337 The sum of the reciprocals of the triangle numbers converge to two. (Contributed by Scott Fenton, 23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |- 
 sum_ k  e.  NN  ( 2  /  (
 k  x.  ( k  +  1 ) ) )  =  2
 
5.8.9  Geometric series
 
Theoremexpcnv 12338* A sequence of powers of a complex number  A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   =>    |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
 
Theoremexplecnv 12339* A sequence of terms converges to zero when it is less than powers of a number  A whose absolute value is smaller than 1. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  <_  ( A ^ k ) )   =>    |-  ( ph  ->  F  ~~>  0 )
 
Theoremgeoserg 12340* The value of the finite geometric series  A ^ M  +  A ^ ( M  + 
1 )  +...  +  A ^
( N  -  1 ). (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  1 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   =>    |-  ( ph  ->  sum_ k  e.  ( M..^ N ) ( A ^ k
 )  =  ( ( ( A ^ M )  -  ( A ^ N ) )  /  ( 1  -  A ) ) )
 
Theoremgeoser 12341* The value of the finite geometric series  1  +  A ^
1  +  A ^
2  +...  +  A ^
( N  -  1 ). (Contributed by NM, 12-May-2006.) (Proof shortened by Mario Carneiro, 15-Jun-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  1 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
 )  =  ( ( 1  -  ( A ^ N ) ) 
 /  ( 1  -  A ) ) )
 
Theoremgeolim 12342* The partial sums in the infinite series  1  +  A ^
1  +  A ^
2... converge to  ( 1  /  (
1  -  A ) ). (Contributed by NM, 15-May-2006.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k
 )  =  ( A ^ k ) )   =>    |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  (
 1  -  A ) ) )
 
Theoremgeolim2 12343* The partial sums in the geometric series  A ^ M  +  A ^ ( M  + 
1 )... converge to  ( ( A ^ M )  / 
( 1  -  A
) ). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  ->  ( F `  k )  =  ( A ^
 k ) )   =>    |-  ( ph  ->  seq 
 M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
 
Theoremgeoreclim 12344* The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  1  <  ( abs `  A ) )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k
 )  =  ( ( 1  /  A ) ^ k ) )   =>    |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( A  /  ( A  -  1 ) ) )
 
Theoremgeo2sum 12345* The value of the finite geometric series  2 ^ -u 1  +  2 ^ -u 2  +...  +  2 ^
-u N, multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  CC )  ->  sum_ k  e.  (
 1 ... N ) ( A  /  ( 2 ^ k ) )  =  ( A  -  ( A  /  (
 2 ^ N ) ) ) )
 
Theoremgeo2sum2 12346* The value of the finite geometric series  1  +  2  + 
4  +  8  +...  +  2 ^ ( N  -  1 ). (Contributed by Mario Carneiro, 7-Sep-2016.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0..^ N ) ( 2 ^ k )  =  ( ( 2 ^ N )  -  1
 ) )
 
Theoremgeo2lim 12347* The value of the infinite geometric series  2 ^ -u 1  +  2 ^ -u 2  +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  F  =  ( k  e.  NN  |->  ( A 
 /  ( 2 ^
 k ) ) )   =>    |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  A )
 
Theoremgeomulcvg 12348* The geometric series converges even if it is multiplied by  k to result in the larger series  k  x.  A ^
k. (Contributed by Mario Carneiro, 27-Mar-2015.)
 |-  F  =  ( k  e.  NN0  |->  ( k  x.  ( A ^
 k ) ) )   =>    |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  F )  e.  dom  ~~>  )
 
Theoremgeoisum 12349* The infinite sum of  1  +  A ^ 1  +  A ^ 2... is  ( 1  /  ( 1  -  A ) ). (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  sum_ k  e.  NN0  ( A ^ k )  =  ( 1  /  (
 1  -  A ) ) )
 
Theoremgeoisumr 12350* The infinite sum of reciprocals  1  +  ( 1  /  A ) ^ 1  +  ( 1  /  A ) ^ 2... is  A  / 
( A  -  1 ). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  1  <  ( abs `  A ) ) 
 ->  sum_ k  e.  NN0  ( ( 1  /  A ) ^ k
 )  =  ( A 
 /  ( A  -  1 ) ) )
 
Theoremgeoisum1 12351* The infinite sum of  A ^ 1  +  A ^ 2... is  ( A  /  ( 1  -  A ) ). (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  sum_ k  e.  NN  ( A ^ k )  =  ( A  /  (
 1  -  A ) ) )
 
Theoremgeoisum1c 12352* The infinite sum of  A  x.  ( R ^ 1 )  +  A  x.  ( R ^ 2 )... is  ( A  x.  R )  /  (
1  -  R ). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( A  x.  ( R ^
 k ) )  =  ( ( A  x.  R )  /  (
 1  -  R ) ) )
 
Theorem0.999... 12353 The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e.  9  /  10 ^
1  +  9  /  10 ^ 2  +  9  /  10 ^ 3  +  ..., is exactly equal to 1, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree. (Contributed by NM, 2-Nov-2007.)
 |- 
 sum_ k  e.  NN  ( 9  /  ( 10 ^ k ) )  =  1
 
Theoremgeoihalfsum 12354 Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 12351. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 12353 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |- 
 sum_ k  e.  NN  ( 1  /  (
 2 ^ k ) )  =  1
 
5.8.10  Ratio test for infinite series convergence
 
Theoremcvgrat 12355* Ratio test for convergence of a complex infinite series. If the ratio  A of the absolute values of successive terms in an infinite sequence  F is less than 1 for all terms beyond some index  B, then the infinite sum of the terms of 
F converges to a complex number. Equivalent to first part of Exercise 4 of [Gleason] p. 182. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 27-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  1 )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  W ) 
 ->  ( abs `  ( F `  ( k  +  1 ) ) ) 
 <_  ( A  x.  ( abs `  ( F `  k ) ) ) )   =>    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
 
5.8.11  Mertens' theorem
 
Theoremmertenslem1 12356* Lemma for mertens 12358. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( ph  /\  j  e.  NN0 )  ->  ( F `  j )  =  A )   &    |-  ( ( ph  /\  j  e.  NN0 )  ->  ( K `  j
 )  =  ( abs `  A ) )   &    |-  (
 ( ph  /\  j  e. 
 NN0 )  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( G `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( H `  k
 )  =  sum_ j  e.  ( 0 ... k
 ) ( A  x.  ( G `  ( k  -  j ) ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  K )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  0 (  +  ,  G )  e.  dom  ~~>  )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  T  =  {
 z  |  E. n  e.  ( 0 ... (
 s  -  1 ) ) z  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
 ) ) }   &    |-  ( ps 
 <->  ( s  e.  NN  /\ 
 A. n  e.  ( ZZ>=
 `  s ) ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
 ) )  <  (
 ( E  /  2
 )  /  ( sum_ j  e.  NN0  ( K `  j )  +  1 ) ) ) )   &    |-  ( ph  ->  ( ps  /\  ( t  e.  NN0  /\ 
 A. m  e.  ( ZZ>=
 `  t ) ( K `  m )  <  ( ( ( E  /  2 ) 
 /  s )  /  ( sup ( T ,  RR ,  <  )  +  1 ) ) ) ) )   &    |-  ( ph  ->  ( 0  <_  sup ( T ,  RR ,  <  ) 
 /\  ( T  C_  RR  /\  T  =/=  (/)  /\  E. z  e.  RR  A. w  e.  T  w  <_  z
 ) ) )   =>    |-  ( ph  ->  E. y  e.  NN0  A. m  e.  ( ZZ>= `  y )
 ( abs `  sum_ j  e.  ( 0 ... m ) ( A  x.  sum_
 k  e.  ( ZZ>= `  ( ( m  -  j )  +  1
 ) ) B ) )  <  E )
 
Theoremmertenslem2 12357* Lemma for mertens 12358. (Contributed by Mario Carneiro, 28-Apr-2014.)
 |-  ( ( ph  /\  j  e.  NN0 )  ->  ( F `  j )  =  A )   &    |-  ( ( ph  /\  j  e.  NN0 )  ->  ( K `  j
 )  =  ( abs `  A ) )   &    |-  (
 ( ph  /\  j  e. 
 NN0 )  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( G `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( H `  k
 )  =  sum_ j  e.  ( 0 ... k
 ) ( A  x.  ( G `  ( k  -  j ) ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  K )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  0 (  +  ,  G )  e.  dom  ~~>  )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  T  =  {
 z  |  E. n  e.  ( 0 ... (
 s  -  1 ) ) z  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
 ) ) }   &    |-  ( ps 
 <->  ( s  e.  NN  /\ 
 A. n  e.  ( ZZ>=
 `  s ) ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
 ) )  <  (
 ( E  /  2
 )  /  ( sum_ j  e.  NN0  ( K `  j )  +  1 ) ) ) )   =>    |-  ( ph  ->  E. y  e.  NN0  A. m  e.  ( ZZ>=
 `  y ) ( abs `  sum_ j  e.  ( 0 ... m ) ( A  x.  sum_
 k  e.  ( ZZ>= `  ( ( m  -  j )  +  1
 ) ) B ) )  <  E )
 
Theoremmertens 12358* Mertens' thoerem. If  A ( j ) is an absolutely convergent series and  B ( k ) is convergent, then  ( sum_ j  e.  NN0 A ( j )  x.  sum_ k  e.  NN0 B ( k ) )  =  sum_ k  e. 
NN0 sum_ j  e.  ( 0 ... k ) ( A ( j )  x.  B ( k  -  j ) ) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( ph  /\  j  e.  NN0 )  ->  ( F `  j )  =  A )   &    |-  ( ( ph  /\  j  e.  NN0 )  ->  ( K `  j
 )  =  ( abs `  A ) )   &    |-  (
 ( ph  /\  j  e. 
 NN0 )  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( G `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( H `  k
 )  =  sum_ j  e.  ( 0 ... k
 ) ( A  x.  ( G `  ( k  -  j ) ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  K )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  0 (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  0
 (  +  ,  H ) 
 ~~>  ( sum_ j  e.  NN0  A  x.  sum_ k  e.  NN0  B ) )
 
5.9  Elementary trigonometry
 
5.9.1  The exponential, sine, and cosine functions
 
Syntaxce 12359 Extend class notation to include the exponential function.
 class  exp
 
Syntaxceu 12360 Extend class notation to include Euler's constant = 2.7182818....
 class  _e
 
Syntaxcsin 12361 Extend class notation to include the sine function.
 class  sin
 
Syntaxccos 12362 Extend class notation to include the cosine function.
 class  cos
 
Syntaxctan 12363 Extend class notation to include the tangent function.
 class  tan
 
Syntaxcpi 12364 Extend class notation to include pi = 3.14159....
 class  pi
 
Definitiondf-ef 12365* Define the exponential function. (Contributed by NM, 14-Mar-2005.)
 |- 
 exp  =  ( x  e.  CC  |->  sum_ k  e.  NN0  ( ( x ^
 k )  /  ( ! `  k ) ) )
 
Definitiondf-e 12366 Define Euler's constant 2.71828.... (Contributed by NM, 14-Mar-2005.)
 |-  _e  =  ( exp `  1 )
 
Definitiondf-sin 12367 Define the sine function. (Contributed by NM, 14-Mar-2005.)
 |- 
 sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) ) 
 /  ( 2  x.  _i ) ) )
 
Definitiondf-cos 12368 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
 |- 
 cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x ) )  +  ( exp `  ( -u _i  x.  x ) ) ) 
 /  2 ) )
 
Definitiondf-tan 12369 Define the tangent function. We define it this way for cmpt 4093, which requires the form  ( x  e.  A  |->  B ). (Contributed by Mario Carneiro, 14-Mar-2014.)
 |- 
 tan  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x ) ) )
 
Definitiondf-pi 12370 Define pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (We use the inverse of less-than, " `'  <", to turn supremum into infimum; currently we don't have infimum defined separately.) (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  pi  =  sup (
 ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )
 
Theoremeftcl 12371 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
 |-  ( ( A  e.  CC  /\  K  e.  NN0 )  ->  ( ( A ^ K )  /  ( ! `  K ) )  e.  CC )
 
Theoremreeftcl 12372 The terms of the series expansion of the exponential function of a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
 |-  ( ( A  e.  RR  /\  K  e.  NN0 )  ->  ( ( A ^ K )  /  ( ! `  K ) )  e.  RR )
 
Theoremeftabs 12373 The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
 |-  ( ( A  e.  CC  /\  K  e.  NN0 )  ->  ( abs `  (
 ( A ^ K )  /  ( ! `  K ) ) )  =  ( ( ( abs `  A ) ^ K )  /  ( ! `  K ) ) )
 
Theoremeftval 12374* The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( N  e.  NN0  ->  ( F `  N )  =  ( ( A ^ N )  /  ( ! `  N ) ) )
 
Theoremefcllem 12375* Lemma for efcl 12380. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 12355 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  seq  0 (  +  ,  F )  e.  dom  ~~>  )
 
Theoremef0lem 12376* The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  =  0  ->  seq  0 (  +  ,  F )  ~~>  1 )
 
Theoremefval 12377* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  CC  ->  ( exp `  A )  =  sum_ k  e. 
 NN0  ( ( A ^ k )  /  ( ! `  k ) ) )
 
Theoremesum 12378 Value of Euler's constant  _e = 2.71828... (Contributed by Steve Rodriguez, 5-Mar-2006.)
 |-  _e  =  sum_ k  e.  NN0  ( 1  /  ( ! `  k ) )
 
Theoremeff 12379 Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
 |- 
 exp : CC --> CC
 
Theoremefcl 12380 Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
 
Theoremefval2 12381* Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  ( exp `  A )  =  sum_ k  e. 
 NN0  ( F `  k ) )
 
Theoremefcvg 12382* The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  seq  0 (  +  ,  F )  ~~>  ( exp `  A ) )
 
Theoremefcvgfsum 12383* Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  sum_ k  e.  ( 0 ... n ) ( ( A ^ k )  /  ( ! `  k ) ) )   =>    |-  ( A  e.  CC  ->  F  ~~>  ( exp `  A ) )
 
Theoremreefcl 12384 The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
 
Theoremreefcld 12385 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( exp `  A )  e. 
 RR )
 
Theoremere 12386 Euler's constant  _e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)
 |-  _e  e.  RR
 
Theoremege2le3 12387 Lemma for egt2lt3 12500. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN  |->  ( 2  x.  ( ( 1 
 /  2 ) ^ n ) ) )   &    |-  G  =  ( n  e.  NN0  |->  ( 1  /  ( ! `  n ) ) )   =>    |-  ( 2  <_  _e  /\  _e  <_  3 )
 
Theoremef0 12388 Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( exp `  0
 )  =  1
 
Theoremefcj 12389 Exponential function of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( A  e.  CC  ->  ( exp `  ( * `  A ) )  =  ( * `  ( exp `  A )
 ) )
 
Theoremefaddlem 12390* Lemma for efadd 12391 (exponential function addition law). (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  G  =  ( n  e.  NN0  |->  ( ( B ^ n )  /  ( ! `  n ) ) )   &    |-  H  =  ( n  e.  NN0  |->  ( ( ( A  +  B ) ^ n )  /  ( ! `  n ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( exp `  ( A  +  B ) )  =  ( ( exp `  A )  x.  ( exp `  B ) ) )
 
Theoremefadd 12391 Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  +  B )
 )  =  ( ( exp `  A )  x.  ( exp `  B ) ) )
 
Theoremefcan 12392 Cancellation of law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
 |-  ( A  e.  CC  ->  ( ( exp `  A )  x.  ( exp `  -u A ) )  =  1
 )
 
Theoremefne0 12393 The exponential function never vanishes. Corollary 15-4.3 of [Gleason] p. 309. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
 
Theoremefneg 12394 Exponent of a negative number. (Contributed by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( exp `  -u A )  =  ( 1  /  ( exp `  A ) ) )
 
Theoremeff2 12395 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
 |- 
 exp : CC --> ( CC  \  { 0 } )
 
Theoremefsub 12396 Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  -  B ) )  =  ( ( exp `  A )  /  ( exp `  B ) ) )
 
Theoremefexp 12397 Exponential function to an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( N  x.  A ) )  =  ( ( exp `  A ) ^ N ) )
 
Theoremefzval 12398 Value of the exponential function for integers. Special case of efval 12377. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( N  e.  ZZ  ->  ( exp `  N )  =  ( _e ^ N ) )
 
Theoremefgt0 12399 The exponential function of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  RR  ->  0  <  ( exp `  A ) )
 
Theoremrpefcl 12400 The exponential function of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR+ )
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