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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiprodclim3 14101* The sequence of partial finite product of a converging infinite product converge to the infinite product of the series. Note that  j must not occur in  A. (Contributed by Scott Fenton, 18-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n (  x.  ,  ( k  e.  Z  |->  A ) )  ~~>  y ) )   &    |-  ( ph  ->  F  e.  dom  ~~>  )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  CC )   &    |-  ( ( ph  /\  j  e.  Z ) 
 ->  ( F `  j
 )  =  prod_ k  e.  ( M ... j
 ) A )   =>    |-  ( ph  ->  F  ~~>  prod_ k  e.  Z  A )
 
Theoremiprodcl 14102* The product of a non-trivially converging infinite sequence is a complex number. (Contributed by Scott Fenton, 18-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y ) )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  Z  A  e.  CC )
 
Theoremiprodrecl 14103* The product of a non-trivially converging infinite real sequence is a real number. (Contributed by Scott Fenton, 18-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y ) )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR )   =>    |-  ( ph  ->  prod_ k  e.  Z  A  e.  RR )
 
Theoremiprodmul 14104* Multiplication of infinite sums. (Contributed by Scott Fenton, 18-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y ) )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  E. m  e.  Z  E. z ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  Z  ( A  x.  B )  =  ( prod_ k  e.  Z  A  x.  prod_ k  e.  Z  B ) )
 
5.10.13  Falling and Rising Factorial
 
Syntaxcfallfac 14105 Declare the syntax for the falling factorial.
 class FallFac
 
Syntaxcrisefac 14106 Declare the syntax for the rising factorial.
 class RiseFac
 
Definitiondf-risefac 14107* Define the rising factorial function. This is the function  ( A  x.  ( A  +  1
)  x.  ... ( A  +  N )
) for complex  A and nonnegative integers  N. (Contributed by Scott Fenton, 5-Jan-2018.)
 |- RiseFac  =  ( x  e.  CC ,  n  e.  NN0  |->  prod_ k  e.  ( 0 ... ( n  -  1 ) ) ( x  +  k
 ) )
 
Definitiondf-fallfac 14108* Define the falling factorial function. This is the function  ( A  x.  ( A  -  1
)  x.  ... ( A  -  N )
) for complex  A and nonnegative integers  N. (Contributed by Scott Fenton, 5-Jan-2018.)
 |- FallFac  =  ( x  e.  CC ,  n  e.  NN0  |->  prod_ k  e.  ( 0 ... ( n  -  1 ) ) ( x  -  k
 ) )
 
Theoremrisefacval 14109* The value of the rising factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  =  prod_ k  e.  (
 0 ... ( N  -  1 ) ) ( A  +  k ) )
 
Theoremfallfacval 14110* The value of the falling factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  =  prod_ k  e.  (
 0 ... ( N  -  1 ) ) ( A  -  k ) )
 
Theoremrisefacval2 14111* One-based value of rising factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  =  prod_ k  e.  (
 1 ... N ) ( A  +  ( k  -  1 ) ) )
 
Theoremfallfacval2 14112* One-based value of falling factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  =  prod_ k  e.  (
 1 ... N ) ( A  -  ( k  -  1 ) ) )
 
Theoremfallfacval3 14113* A product representation of falling factorial when  A is a nonnegative integer. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( N  e.  (
 0 ... A )  ->  ( A FallFac  N )  = 
 prod_ k  e.  (
 ( A  -  ( N  -  1 ) )
 ... A ) k )
 
Theoremrisefaccllem 14114* Lemma for rising factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  S  C_  CC   &    |-  1  e.  S   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  x.  y )  e.  S )   &    |-  ( ( A  e.  S  /\  k  e.  NN0 )  ->  ( A  +  k )  e.  S )   =>    |-  ( ( A  e.  S  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  S )
 
Theoremfallfaccllem 14115* Lemma for falling factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  S  C_  CC   &    |-  1  e.  S   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  x.  y )  e.  S )   &    |-  ( ( A  e.  S  /\  k  e.  NN0 )  ->  ( A  -  k )  e.  S )   =>    |-  ( ( A  e.  S  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  S )
 
Theoremrisefaccl 14116 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  CC )
 
Theoremfallfaccl 14117 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  CC )
 
Theoremrerisefaccl 14118 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  RR )
 
Theoremrefallfaccl 14119 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  RR )
 
Theoremnnrisefaccl 14120 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  NN  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  NN )
 
Theoremzrisefaccl 14121 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  ZZ )
 
Theoremzfallfaccl 14122 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  ZZ )
 
Theoremnn0risefaccl 14123 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  NN0  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  NN0 )
 
Theoremrprisefaccl 14124 Closure law for rising factorial. (Contributed by Scott Fenton, 9-Jan-2018.)
 |-  ( ( A  e.  RR+  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  RR+ )
 
Theoremrisefallfac 14125 A relationship between rising and falling factorials. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  ( X RiseFac  N )  =  ( ( -u 1 ^ N )  x.  ( -u X FallFac  N )
 ) )
 
Theoremfallrisefac 14126 A relationship between falling and rising factorials. (Contributed by Scott Fenton, 17-Jan-2018.)
 |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  ( X FallFac  N )  =  ( ( -u 1 ^ N )  x.  ( -u X RiseFac  N )
 ) )
 
Theoremrisefall0lem 14127 Lemma for risefac0 14128 and fallfac0 14129. Show a particular set of finite integers is empty. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( 0 ... (
 0  -  1 ) )  =  (/)
 
Theoremrisefac0 14128 The value of the rising factorial when  N  =  0. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A RiseFac  0 )  =  1 )
 
Theoremfallfac0 14129 The value of the falling factorial when  N  =  0. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A FallFac  0 )  =  1 )
 
Theoremrisefacp1 14130 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  ( N  +  1 )
 )  =  ( ( A RiseFac  N )  x.  ( A  +  N )
 ) )
 
Theoremfallfacp1 14131 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  ( N  +  1 )
 )  =  ( ( A FallFac  N )  x.  ( A  -  N ) ) )
 
Theoremrisefacp1d 14132 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A RiseFac  ( N  +  1 ) )  =  ( ( A RiseFac  N )  x.  ( A  +  N ) ) )
 
Theoremfallfacp1d 14133 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A FallFac  ( N  +  1 ) )  =  ( ( A FallFac  N )  x.  ( A  -  N ) ) )
 
Theoremrisefac1 14134 The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A RiseFac  1 )  =  A )
 
Theoremfallfac1 14135 The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A FallFac  1 )  =  A )
 
Theoremrisefacfac 14136 Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( N  e.  NN0  ->  ( 1 RiseFac  N )  =  ( ! `  N ) )
 
Theoremfallfacfwd 14137 The forward difference of a falling factorial. (Contributed by Scott Fenton, 21-Jan-2018.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( ( A  +  1 ) FallFac  N )  -  ( A FallFac  N ) )  =  ( N  x.  ( A FallFac  ( N  -  1
 ) ) ) )
 
Theorem0fallfac 14138 The value of the zero falling factorial at natural  N. (Contributed by Scott Fenton, 17-Feb-2018.)
 |-  ( N  e.  NN  ->  ( 0 FallFac  N )  =  0 )
 
Theorem0risefac 14139 The value of the zero rising factorial at natural  N. (Contributed by Scott Fenton, 17-Feb-2018.)
 |-  ( N  e.  NN  ->  ( 0 RiseFac  N )  =  0 )
 
Theorembinomfallfaclem1 14140 Lemma for binomfallfac 14142. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ( ph  /\  K  e.  ( 0 ... N ) )  ->  ( ( N  _C  K )  x.  ( ( A FallFac  ( N  -  K ) )  x.  ( B FallFac  ( K  +  1 ) ) ) )  e.  CC )
 
Theorembinomfallfaclem2 14141* Lemma for binomfallfac 14142. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ps  ->  (
 ( A  +  B ) FallFac  N )  =  sum_ k  e.  ( 0 ...
 N ) ( ( N  _C  k )  x.  ( ( A FallFac  ( N  -  k
 ) )  x.  ( B FallFac  k ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( A  +  B ) FallFac  ( N  +  1 )
 )  =  sum_ k  e.  ( 0 ... ( N  +  1 )
 ) ( ( ( N  +  1 )  _C  k )  x.  ( ( A FallFac  (
 ( N  +  1 )  -  k ) )  x.  ( B FallFac  k ) ) ) )
 
Theorembinomfallfac 14142* A version of the binomial theorem using falling factorials instead of exponentials. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) FallFac  N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A FallFac  ( N  -  k ) )  x.  ( B FallFac  k )
 ) ) )
 
Theorembinomrisefac 14143* A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) RiseFac  N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A RiseFac  ( N  -  k ) )  x.  ( B RiseFac  k )
 ) ) )
 
Theoremfallfacval4 14144 Represent the falling factorial via factorials when the first argument is a natural. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( N  e.  (
 0 ... A )  ->  ( A FallFac  N )  =  ( ( ! `  A )  /  ( ! `  ( A  -  N ) ) ) )
 
Theorembcfallfac 14145 Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  =  ( ( N FallFac  K )  /  ( ! `  K ) ) )
 
Theoremfallfacfac 14146 Relate falling factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( N  e.  NN0  ->  ( N FallFac  N )  =  ( ! `  N ) )
 
5.10.14  Bernoulli polynomials and sums of k-th powers
 
Syntaxcbp 14147 Declare the constant for the Bernoulli polynomial operator.
 class BernPoly
 
Definitiondf-bpoly 14148* Define the Bernoulli polynomials. Here we use well-founded recursion to define the Bernoulli polynomials. This agrees with most textbook definitions, although explicit formulae do exist. (Contributed by Scott Fenton, 22-May-2014.)
 |- BernPoly  =  ( m  e.  NN0 ,  x  e.  CC  |->  (wrecs (  <  ,  NN0 ,  ( g  e.  _V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( x ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `
  k )  /  ( ( n  -  k )  +  1
 ) ) ) ) ) ) `  m ) )
 
Theorembpolylem 14149* Lemma for bpolyval 14150. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  G  =  ( g  e.  _V  |->  [_ ( # `
  dom  g )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
 ( n  -  k
 )  +  1 ) ) ) ) )   &    |-  F  = wrecs (  <  , 
 NN0 ,  G )   =>    |-  (
 ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
 ( k BernPoly  X )  /  ( ( N  -  k )  +  1
 ) ) ) ) )
 
Theorembpolyval 14150* The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
 ( k BernPoly  X )  /  ( ( N  -  k )  +  1
 ) ) ) ) )
 
Theorembpoly0 14151 The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( X  e.  CC  ->  ( 0 BernPoly  X )  =  1 )
 
Theorembpoly1 14152 The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( X  e.  CC  ->  ( 1 BernPoly  X )  =  ( X  -  ( 1  /  2
 ) ) )
 
Theorembpolycl 14153 Closure law for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  e.  CC )
 
Theorembpolysum 14154* A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  ( ( N  e.  NN0  /\  X  e.  CC )  -> 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( k BernPoly  X )  /  (
 ( N  -  k
 )  +  1 ) ) )  =  ( X ^ N ) )
 
Theorembpolydiflem 14155* Lemma for bpolydif 14156. (Contributed by Scott Fenton, 12-Jun-2014.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  X  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( (
 k BernPoly  ( X  +  1 ) )  -  (
 k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  -  1 ) ) ) )   =>    |-  ( ph  ->  (
 ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^ ( N  -  1 ) ) ) )
 
Theorembpolydif 14156 Calculate the difference between successive values of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 26-May-2014.)
 |-  ( ( N  e.  NN  /\  X  e.  CC )  ->  ( ( N BernPoly  ( X  +  1
 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^ ( N  -  1 ) ) ) )
 
Theoremfsumkthpow 14157* A closed-form expression for the sum of  K-th powers. (Contributed by Scott Fenton, 16-May-2014.) This is Metamath 100 proof #77. (Revised by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
 sum_ n  e.  (
 0 ... M ) ( n ^ K )  =  ( ( ( ( K  +  1 ) BernPoly  ( M  +  1 ) )  -  (
 ( K  +  1 ) BernPoly  0 ) ) 
 /  ( K  +  1 ) ) )
 
Theorembpoly2 14158 The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 2 BernPoly  X )  =  ( ( ( X ^ 2 )  -  X )  +  ( 1  /  6
 ) ) )
 
Theorembpoly3 14159 The Bernoulli polynomials at three. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 3 BernPoly  X )  =  ( ( ( X ^ 3 )  -  ( ( 3 
 /  2 )  x.  ( X ^ 2
 ) ) )  +  ( ( 1  / 
 2 )  x.  X ) ) )
 
Theorembpoly4 14160 The Bernoulli polynomials at four. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 4 BernPoly  X )  =  ( ( ( ( X ^ 4
 )  -  ( 2  x.  ( X ^
 3 ) ) )  +  ( X ^
 2 ) )  -  ( 1  / ; 3 0 ) ) )
 
Theoremfsumcube 14161* Express the sum of cubes in closed terms. (Contributed by Scott Fenton, 16-Jun-2015.)
 |-  ( T  e.  NN0  ->  sum_ k  e.  ( 0
 ... T ) ( k ^ 3 )  =  ( ( ( T ^ 2 )  x.  ( ( T  +  1 ) ^
 2 ) )  / 
 4 ) )
 
5.11  Elementary trigonometry
 
5.11.1  The exponential, sine, and cosine functions
 
Syntaxce 14162 Extend class notation to include the exponential function.
 class  exp
 
Syntaxceu 14163 Extend class notation to include Euler's constant = 2.7182818....
 class  _e
 
Syntaxcsin 14164 Extend class notation to include the sine function.
 class  sin
 
Syntaxccos 14165 Extend class notation to include the cosine function.
 class  cos
 
Syntaxctan 14166 Extend class notation to include the tangent function.
 class  tan
 
Syntaxcpi 14167 Extend class notation to include pi = 3.14159....
 class  pi
 
Syntaxcpiold 14168 Extend class notation to include pi = 3.14159.... (old version)
 class  pi
 
Definitiondf-ef 14169* Define the exponential function. (Contributed by NM, 14-Mar-2005.)
 |- 
 exp  =  ( x  e.  CC  |->  sum_ k  e.  NN0  ( ( x ^
 k )  /  ( ! `  k ) ) )
 
Definitiondf-e 14170 Define Euler's constant 2.71828.... (Contributed by NM, 14-Mar-2005.)
 |-  _e  =  ( exp `  1 )
 
Definitiondf-sin 14171 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 14172 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 14173 Define the tangent function. We define it this way for cmpt 4474, 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 14174 Define pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
 |-  pi  = inf ( (
 RR+  i^i  ( `' sin " { 0 } )
 ) ,  RR ,  <  )
 
Definitiondf-piOLD 14175 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.) Obsolete version of df-pi 14174 as of 14-Sep-2020. (New usage is discouraged.)
 |-  pi  =  sup (
 ( RR+  i^i  ( `' sin " { 0 } ) ) ,  RR ,  `'  <  )
 
Theoremeftcl 14176 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 14177 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 14178 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 14179* 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 14180* Lemma for efcl 14185. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 13987 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 14181* 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 14182* 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 14183 Value of Euler's constant  _e = 2.71828... (Contributed by Steve Rodriguez, 5-Mar-2006.)
 |-  _e  =  sum_ k  e.  NN0  ( 1  /  ( ! `  k ) )
 
Theoremeff 14184 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 14185 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 14186* 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 14187* 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 14188* 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 14189 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 14190 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 14191 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 14192 Lemma for egt2lt3 14306. (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 14193 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 14194 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 14195* Lemma for efadd 14196 (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 14196 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 ) ) )
 
Theoremfprodefsum 14197* Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  ( M ... N ) ( exp `  A )  =  ( exp ` 
 sum_ k  e.  ( M ... N ) A ) )
 
Theoremefcan 14198 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 14199 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 14200 Exponent of a negative number. (Contributed by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( exp `  -u A )  =  ( 1  /  ( exp `  A ) ) )
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