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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcxploglim 23901* The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( ( log `  n )  /  ( n  ^c  A ) ) )  ~~> r  0 )
 
Theoremcxploglim2 23902* Every power of the logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 20-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  RR+ )  ->  ( n  e.  RR+  |->  ( ( ( log `  n )  ^c  A )  /  ( n  ^c  B ) ) )  ~~> r  0 )
 
Theoremdivsqrtsumlem 23903* Lemma for divsqrsum 23905 and divsqrtsum2 23906. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   =>    |-  ( F : RR+ --> RR 
 /\  F  e.  dom  ~~> r 
 /\  ( ( F  ~~> r  L  /\  A  e.  RR+ )  ->  ( abs `  ( ( F `  A )  -  L ) )  <_  ( 1 
 /  ( sqr `  A ) ) ) )
 
Theoremdivsqrsumf 23904* The function  F used in divsqrsum 23905 is a real function. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   =>    |-  F : RR+ --> RR
 
Theoremdivsqrsum 23905* The sum  sum_ n  <_  x ( 1  /  sqr n ) is asymptotic to  2 sqr x  +  L with a finite limit  L. (In fact, this limit is  zeta ( 1  /  2 )  ~~  -u 1 period 4 6 ....) (Contributed by Mario Carneiro, 9-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   =>    |-  F  e.  dom  ~~> r
 
Theoremdivsqrtsum2 23906* A bound on the distance of the sum  sum_ n  <_  x (
1  /  sqr n
) from its asymptotic value  2 sqr x  +  L. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   &    |-  ( ph  ->  F  ~~> r  L )   =>    |-  ( ( ph  /\  A  e.  RR+ )  ->  ( abs `  ( ( F `
  A )  -  L ) )  <_  ( 1  /  ( sqr `  A ) ) )
 
Theoremdivsqrtsumo1 23907* The sum  sum_ n  <_  x ( 1  /  sqr n ) has the asymptotic expansion  2 sqr x  +  L  +  O
( 1  /  sqr x ), for some  L. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   &    |-  ( ph  ->  F  ~~> r  L )   =>    |-  ( ph  ->  (
 y  e.  RR+  |->  ( ( ( F `  y
 )  -  L )  x.  ( sqr `  y
 ) ) )  e.  O(1) )
 
14.3.12  Inequality of arithmetic and geometric means
 
Theoremcvxcl 23908* Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D ) )  ->  ( x [,] y )  C_  D )   =>    |-  ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) ) 
 ->  ( ( T  x.  X )  +  (
 ( 1  -  T )  x.  Y ) )  e.  D )
 
Theoremscvxcvx 23909* A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  ( a  e.  D  /\  b  e.  D ) )  ->  ( a [,] b
 )  C_  D )   &    |-  (
 ( ph  /\  ( x  e.  D  /\  y  e.  D  /\  x  < 
 y )  /\  t  e.  ( 0 (,) 1
 ) )  ->  ( F `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
 ) ) )  < 
 ( ( t  x.  ( F `  x ) )  +  (
 ( 1  -  t
 )  x.  ( F `
  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) ) 
 ->  ( F `  (
 ( T  x.  X )  +  ( (
 1  -  T )  x.  Y ) ) )  <_  ( ( T  x.  ( F `  X ) )  +  ( ( 1  -  T )  x.  ( F `  Y ) ) ) )
 
Theoremjensenlem1 23910* Lemma for jensen 23912. (Contributed by Mario Carneiro, 4-Jun-2016.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  ( a  e.  D  /\  b  e.  D ) )  ->  ( a [,] b
 )  C_  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  T : A --> ( 0 [,) +oo ) )   &    |-  ( ph  ->  X : A --> D )   &    |-  ( ph  ->  0  <  (fld  gsumg  T ) )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D  /\  t  e.  (
 0 [,] 1 ) ) )  ->  ( F `  ( ( t  x.  x )  +  (
 ( 1  -  t
 )  x.  y ) ) )  <_  (
 ( t  x.  ( F `  x ) )  +  ( ( 1  -  t )  x.  ( F `  y
 ) ) ) )   &    |-  ( ph  ->  -.  z  e.  B )   &    |-  ( ph  ->  ( B  u.  { z } )  C_  A )   &    |-  S  =  (fld  gsumg  ( T  |`  B ) )   &    |-  L  =  (fld  gsumg  ( T  |`  ( B  u.  {
 z } ) ) )   =>    |-  ( ph  ->  L  =  ( S  +  ( T `  z ) ) )
 
Theoremjensenlem2 23911* Lemma for jensen 23912. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  ( a  e.  D  /\  b  e.  D ) )  ->  ( a [,] b
 )  C_  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  T : A --> ( 0 [,) +oo ) )   &    |-  ( ph  ->  X : A --> D )   &    |-  ( ph  ->  0  <  (fld  gsumg  T ) )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D  /\  t  e.  (
 0 [,] 1 ) ) )  ->  ( F `  ( ( t  x.  x )  +  (
 ( 1  -  t
 )  x.  y ) ) )  <_  (
 ( t  x.  ( F `  x ) )  +  ( ( 1  -  t )  x.  ( F `  y
 ) ) ) )   &    |-  ( ph  ->  -.  z  e.  B )   &    |-  ( ph  ->  ( B  u.  { z } )  C_  A )   &    |-  S  =  (fld  gsumg  ( T  |`  B ) )   &    |-  L  =  (fld  gsumg  ( T  |`  ( B  u.  {
 z } ) ) )   &    |-  ( ph  ->  S  e.  RR+ )   &    |-  ( ph  ->  ( (fld 
 gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D )   &    |-  ( ph  ->  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  <_  ( (fld 
 gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )   =>    |-  ( ph  ->  ( ( (fld 
 gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
 z } ) ) )  /  L )  e.  D  /\  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
 z } ) ) )  /  L ) )  <_  ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) ) 
 /  L ) ) )
 
Theoremjensen 23912* Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (Contributed by Mario Carneiro, 21-Jun-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  ( a  e.  D  /\  b  e.  D ) )  ->  ( a [,] b
 )  C_  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  T : A --> ( 0 [,) +oo ) )   &    |-  ( ph  ->  X : A --> D )   &    |-  ( ph  ->  0  <  (fld  gsumg  T ) )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D  /\  t  e.  (
 0 [,] 1 ) ) )  ->  ( F `  ( ( t  x.  x )  +  (
 ( 1  -  t
 )  x.  y ) ) )  <_  (
 ( t  x.  ( F `  x ) )  +  ( ( 1  -  t )  x.  ( F `  y
 ) ) ) )   =>    |-  ( ph  ->  ( (
 (fld  gsumg  ( T  oF  x.  X ) )  /  (fld  gsumg  T ) )  e.  D  /\  ( F `  (
 (fld  gsumg  ( T  oF  x.  X ) )  /  (fld  gsumg  T ) ) )  <_  ( (fld 
 gsumg  ( T  oF  x.  ( F  o.  X ) ) )  /  (fld  gsumg  T ) ) ) )
 
Theoremamgmlem 23913 Lemma for amgm 23914. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  M  =  (mulGrp ` fld )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  F : A --> RR+ )   =>    |-  ( ph  ->  (
 ( M  gsumg 
 F )  ^c 
 ( 1  /  ( # `
  A ) ) )  <_  ( (fld  gsumg 
 F )  /  ( # `
  A ) ) )
 
Theoremamgm 23914 Inequality of arithmetic and geometric means. Here  ( M  gsumg  F ) calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements  F ( x ) ,  x  e.  A together), and  (fld 
gsumg  F ) calculates the group sum in the additive group (i.e. the sum of the elements). This is Metamath 100 proof #38. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  M  =  (mulGrp ` fld )   =>    |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,) +oo ) )  ->  (
 ( M  gsumg 
 F )  ^c 
 ( 1  /  ( # `
  A ) ) )  <_  ( (fld  gsumg 
 F )  /  ( # `
  A ) ) )
 
14.3.13  Euler-Mascheroni constant
 
Syntaxcem 23915 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)
 class  gamma
 
Definitiondf-em 23916 Define the Euler-Mascheroni constant,  gamma  = 0.577... . This is the limit of the series  sum_ k  e.  ( 1 ... m
) ( 1  / 
k )  -  ( log `  m ), with a proof that the limit exists in emcl 23926. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  =  sum_ k  e. 
 NN  ( ( 1 
 /  k )  -  ( log `  ( 1  +  ( 1  /  k
 ) ) ) )
 
Theoremlogdifbnd 23917 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)
 |-  ( A  e.  RR+  ->  ( ( log `  ( A  +  1 )
 )  -  ( log `  A ) )  <_  ( 1  /  A ) )
 
Theoremlogdiflbnd 23918 Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( A  e.  RR+  ->  ( 1  /  ( A  +  1 )
 )  <_  ( ( log `  ( A  +  1 ) )  -  ( log `  A )
 ) )
 
Theorememcllem1 23919* Lemma for emcl 23926. The series  F and 
G are sequences of real numbers that approach 
gamma from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   =>    |-  ( F : NN
 --> RR  /\  G : NN
 --> RR )
 
Theorememcllem2 23920* Lemma for emcl 23926. 
F is increasing, and  G is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   =>    |-  ( N  e.  NN  ->  ( ( F `
  ( N  +  1 ) )  <_  ( F `  N ) 
 /\  ( G `  N )  <_  ( G `
  ( N  +  1 ) ) ) )
 
Theorememcllem3 23921* Lemma for emcl 23926. The function  H is the difference between  F and  G. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   =>    |-  ( N  e.  NN  ->  ( H `  N )  =  ( ( F `  N )  -  ( G `  N ) ) )
 
Theorememcllem4 23922* Lemma for emcl 23926. The difference between series  F and  G tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   =>    |-  H 
 ~~>  0
 
Theorememcllem5 23923* Lemma for emcl 23926. The partial sums of the series  T, which is used in the definition df-em 23916, is in fact the same as  G. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   &    |-  T  =  ( n  e.  NN  |->  ( ( 1 
 /  n )  -  ( log `  ( 1  +  ( 1  /  n ) ) ) ) )   =>    |-  G  =  seq 1
 (  +  ,  T )
 
Theorememcllem6 23924* Lemma for emcl 23926. By the previous lemmas,  F and  G must approach a common limit, which is  gamma by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   &    |-  T  =  ( n  e.  NN  |->  ( ( 1 
 /  n )  -  ( log `  ( 1  +  ( 1  /  n ) ) ) ) )   =>    |-  ( F  ~~>  gamma  /\  G  ~~>  gamma
 )
 
Theorememcllem7 23925* Lemma for emcl 23926 and harmonicbnd 23927. Derive bounds on  gamma as  F ( 1 ) and  G ( 1 ). (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   &    |-  T  =  ( n  e.  NN  |->  ( ( 1 
 /  n )  -  ( log `  ( 1  +  ( 1  /  n ) ) ) ) )   =>    |-  ( gamma  e.  (
 ( 1  -  ( log `  2 ) ) [,] 1 )  /\  F : NN --> ( gamma [,] 1 )  /\  G : NN --> ( ( 1  -  ( log `  2
 ) ) [,] gamma ) )
 
Theorememcl 23926 Closure and bounds for the Euler-Mascheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  e.  ( ( 1  -  ( log `  2
 ) ) [,] 1
 )
 
Theoremharmonicbnd 23927* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( N  e.  NN  ->  ( sum_ m  e.  (
 1 ... N ) ( 1  /  m )  -  ( log `  N ) )  e.  ( gamma [,] 1 ) )
 
Theoremharmonicbnd2 23928* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( N  e.  NN  ->  ( sum_ m  e.  (
 1 ... N ) ( 1  /  m )  -  ( log `  ( N  +  1 )
 ) )  e.  (
 ( 1  -  ( log `  2 ) ) [,] gamma ) )
 
Theorememre 23929 The Euler-Mascheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  e.  RR
 
Theorememgt0 23930 The Euler-Mascheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  0  <  gamma
 
Theoremharmonicbnd3 23931* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( N  e.  NN0  ->  ( sum_ m  e.  (
 1 ... N ) ( 1  /  m )  -  ( log `  ( N  +  1 )
 ) )  e.  (
 0 [,] gamma ) )
 
Theoremharmoniclbnd 23932* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( A  e.  RR+  ->  ( log `  A )  <_ 
 sum_ m  e.  (
 1 ... ( |_ `  A ) ) ( 1 
 /  m ) )
 
Theoremharmonicubnd 23933* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ m  e.  (
 1 ... ( |_ `  A ) ) ( 1 
 /  m )  <_  ( ( log `  A )  +  1 )
 )
 
Theoremharmonicbnd4 23934* The asymptotic behavior of  sum_ m  <_  A ,  1  /  m  =  log A  +  gamma  +  O ( 1  /  A ). (Contributed by Mario Carneiro, 14-May-2016.)
 |-  ( A  e.  RR+  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1 
 /  m )  -  ( ( log `  A )  +  gamma ) ) )  <_  ( 1  /  A ) )
 
Theoremfsumharmonic 23935* Bound a finite sum based on the harmonic series, where the "strong" bound  C only applies asymptotically, and there is a "weak" bound  R for the remaining values. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  ( T  e.  RR  /\  1  <_  T ) )   &    |-  ( ph  ->  ( R  e.  RR  /\  0  <_  R ) )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  B  e.  CC )   &    |-  ( ( ph  /\  n  e.  ( 1
 ... ( |_ `  A ) ) )  ->  C  e.  RR )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <_  C )   &    |-  ( ( (
 ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  T  <_  ( A  /  n ) )  ->  ( abs `  B )  <_  ( C  x.  n ) )   &    |-  ( ( ( ph  /\  n  e.  ( 1
 ... ( |_ `  A ) ) )  /\  ( A  /  n )  <  T )  ->  ( abs `  B )  <_  R )   =>    |-  ( ph  ->  ( abs `  sum_ n  e.  (
 1 ... ( |_ `  A ) ) ( B 
 /  n ) ) 
 <_  ( sum_ n  e.  (
 1 ... ( |_ `  A ) ) C  +  ( R  x.  (
 ( log `  T )  +  1 ) ) ) )
 
14.3.14  Zeta function
 
Syntaxczeta 23936 The Riemann zeta function.
 class  zeta
 
Definitiondf-zeta 23937* Define the Riemann zeta function. This definition uses a series expansion of the alternating zeta function ~? zetaalt that is convergent everywhere except  1, but going from the alternating zeta function to the regular zeta function requires dividing by  1  -  2 ^ ( 1  -  s ), which has zeroes other than  1. To extract the correct value of the zeta function at these points, we extend the divided alternating zeta function by continuity. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |- 
 zeta  =  ( iota_ f  e.  ( ( CC  \  { 1 } ) -cn->
 CC ) A. s  e.  ( CC  \  {
 1 } ) ( ( 1  -  (
 2  ^c  ( 1  -  s ) ) )  x.  (
 f `  s )
 )  =  sum_ n  e.  NN0  ( sum_ k  e.  ( 0 ... n ) ( ( (
 -u 1 ^ k
 )  x.  ( n  _C  k ) )  x.  ( ( k  +  1 )  ^c  s ) ) 
 /  ( 2 ^
 ( n  +  1 ) ) ) )
 
Theoremzetacvg 23938* The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ph  ->  S  e.  CC )   &    |-  ( ph  ->  1  <  ( Re `  S ) )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  k )  =  ( k  ^c  -u S ) )   =>    |-  ( ph  ->  seq 1 (  +  ,  F )  e.  dom  ~~>  )
 
14.3.15  Gamma function
 
Syntaxclgam 23939 Logarithm of the Gamma function.
 class  log _G
 
Syntaxcgam 23940 The Gamma function.
 class  _G
 
Syntaxcigam 23941 The inverse Gamma function.
 class 1/ _G
 
Definitiondf-lgam 23942* Define the log-Gamma function. We can work with this form of the gamma function a bit easier than the equivalent expression for the gamma function itself, and moreover this function is not actually equal to  log ( _G ( x ) ) because the branch cuts are placed differently (we do have  exp ( log _G ( x ) )  =  _G ( x ), though). This definition is attributed to Euler, and unlike the usual integral definition is defined on the entire complex plane except the nonpositive integers  ZZ  \  NN, where the function has simple poles. (Contributed by Mario Carneiro, 12-Jul-2014.)
 |- 
 log _G  =  ( z  e.  ( CC  \  ( ZZ  \  NN )
 )  |->  ( sum_ m  e.  NN  ( ( z  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( z 
 /  m )  +  1 ) ) )  -  ( log `  z
 ) ) )
 
Definitiondf-gam 23943 Define the Gamma function. See df-lgam 23942 for more information about the reason for this definition in terms of the log-gamma function. (Contributed by Mario Carneiro, 12-Jul-2014.)
 |- 
 _G  =  ( exp 
 o.  log _G )
 
Definitiondf-igam 23944 Define the inverse Gamma function, which is defined everywhere, unlike the Gamma function itself. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |- 1/ _G  =  ( x  e.  CC  |->  if ( x  e.  ( ZZ  \  NN ) ,  0 ,  ( 1  /  ( _G `  x ) ) ) )
 
Theoremeldmgm 23945 Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  <->  ( A  e.  CC  /\  -.  -u A  e.  NN0 ) )
 
Theoremdmgmaddn0 23946 If  A is not a nonpositive integer, then  A  +  N is nonzero for any nonnegative integer  N. (Contributed by Mario Carneiro, 12-Jul-2014.)
 |-  ( ( A  e.  ( CC  \  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  ( A  +  N )  =/=  0 )
 
Theoremdmlogdmgm 23947 If  A is in the continuous domain of the logarithm, then it is in the domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( -oo (,] 0 ) )  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )
 
Theoremrpdmgm 23948 A positive real number is in the domain of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  RR+  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )
 
Theoremdmgmn0 23949 If  A is not a nonpositive integer, then  A is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremdmgmaddnn0 23950 If  A is not a nonpositive integer and  N is a nonnegative integer, then  A  +  N is also not a nonpositive integer. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A  +  N )  e.  ( CC  \  ( ZZ  \  NN ) ) )
 
Theoremdmgmdivn0 23951 Lemma for lgamf 23965. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   &    |-  ( ph  ->  M  e.  NN )   =>    |-  ( ph  ->  ( ( A  /  M )  +  1 )  =/=  0 )
 
Theoremlgamgulmlem1 23952* Lemma for lgamgulm 23958. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   =>    |-  ( ph  ->  U  C_  ( CC  \  ( ZZ  \  NN ) ) )
 
Theoremlgamgulmlem2 23953* Lemma for lgamgulm 23958. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  (
 2  x.  R ) 
 <_  N )   =>    |-  ( ph  ->  ( abs `  ( ( A 
 /  N )  -  ( log `  ( ( A  /  N )  +  1 ) ) ) )  <_  ( R  x.  ( ( 1  /  ( N  -  R ) )  -  (
 1  /  N )
 ) ) )
 
Theoremlgamgulmlem3 23954* Lemma for lgamgulm 23958. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  (
 2  x.  R ) 
 <_  N )   =>    |-  ( ph  ->  ( abs `  ( ( A  x.  ( log `  (
 ( N  +  1 )  /  N ) ) )  -  ( log `  ( ( A 
 /  N )  +  1 ) ) ) )  <_  ( R  x.  ( ( 2  x.  ( R  +  1 ) )  /  ( N ^ 2 ) ) ) )
 
Theoremlgamgulmlem4 23955* Lemma for lgamgulm 23958. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( z  /  m )  +  1 )
 ) ) ) )   &    |-  T  =  ( m  e.  NN  |->  if ( ( 2  x.  R )  <_  m ,  ( R  x.  ( ( 2  x.  ( R  +  1 ) )  /  ( m ^ 2 ) ) ) ,  ( ( R  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  +  ( ( log `  (
 ( R  +  1 )  x.  m ) )  +  pi ) ) ) )   =>    |-  ( ph  ->  seq 1 (  +  ,  T )  e.  dom  ~~>  )
 
Theoremlgamgulmlem5 23956* Lemma for lgamgulm 23958. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( z  /  m )  +  1 )
 ) ) ) )   &    |-  T  =  ( m  e.  NN  |->  if ( ( 2  x.  R )  <_  m ,  ( R  x.  ( ( 2  x.  ( R  +  1 ) )  /  ( m ^ 2 ) ) ) ,  ( ( R  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  +  ( ( log `  (
 ( R  +  1 )  x.  m ) )  +  pi ) ) ) )   =>    |-  ( ( ph  /\  ( n  e.  NN  /\  y  e.  U ) )  ->  ( abs `  ( ( G `  n ) `  y
 ) )  <_  ( T `  n ) )
 
Theoremlgamgulmlem6 23957* The series  G is uniformly convergent on the compact region  U, which describes a circle of radius  R with holes of size  1  /  R around the poles of the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( z  /  m )  +  1 )
 ) ) ) )   &    |-  T  =  ( m  e.  NN  |->  if ( ( 2  x.  R )  <_  m ,  ( R  x.  ( ( 2  x.  ( R  +  1 ) )  /  ( m ^ 2 ) ) ) ,  ( ( R  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  +  ( ( log `  (
 ( R  +  1 )  x.  m ) )  +  pi ) ) ) )   =>    |-  ( ph  ->  ( 
 seq 1 (  oF  +  ,  G )  e.  dom  ( ~~> u `  U )  /\  (  seq 1 (  oF  +  ,  G )
 ( ~~> u `  U ) ( z  e.  U  |->  O )  ->  E. r  e.  RR  A. z  e.  U  ( abs `  O )  <_  r ) ) )
 
Theoremlgamgulm 23958* The series  G is uniformly convergent on the compact region  U, which describes a circle of radius  R with holes of size 
1  /  R around the poles of the gamma function. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( z  /  m )  +  1 )
 ) ) ) )   =>    |-  ( ph  ->  seq 1
 (  oF  +  ,  G )  e.  dom  (
 ~~> u `  U ) )
 
Theoremlgamgulm2 23959* Rewrite the limit of the sequence 
G in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( z  /  m )  +  1 )
 ) ) ) )   =>    |-  ( ph  ->  ( A. z  e.  U  ( log _G `  z )  e.  CC  /\  seq 1 (  oF  +  ,  G )
 ( ~~> u `  U ) ( z  e.  U  |->  ( ( log _G `  z )  +  ( log `  z )
 ) ) ) )
 
Theoremlgambdd 23960* The log-Gamma function is bounded on the region  U. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1 
 /  R )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( z  /  m )  +  1 )
 ) ) ) )   =>    |-  ( ph  ->  E. r  e.  RR  A. z  e.  U  ( abs `  ( log _G `  z ) )  <_  r )
 
Theoremlgamucov 23961* The  U regions used in the proof of lgamgulm 23958 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  r  /\  A. k  e.  NN0  ( 1  /  r )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   &    |-  J  =  ( TopOpen ` fld )   =>    |-  ( ph  ->  E. r  e.  NN  A  e.  (
 ( int `  J ) `  U ) )
 
Theoremlgamucov2 23962* The  U regions used in the proof of lgamgulm 23958 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  r  /\  A. k  e.  NN0  ( 1  /  r )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  E. r  e.  NN  A  e.  U )
 
Theoremlgamcvglem 23963* Lemma for lgamf 23965 and lgamcvg 23977. (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  r  /\  A. k  e.  NN0  ( 1  /  r )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( A  /  m )  +  1 )
 ) ) )   =>    |-  ( ph  ->  ( ( log _G `  A )  e.  CC  /\  seq 1 (  +  ,  G ) 
 ~~>  ( ( log _G `  A )  +  ( log `  A ) ) ) )
 
Theoremlgamcl 23964 The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( log _G `  A )  e.  CC )
 
Theoremlgamf 23965 The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |- 
 log _G : ( CC  \  ( ZZ  \  NN ) ) --> CC
 
Theoremgamf 23966 The Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |- 
 _G : ( CC  \  ( ZZ  \  NN ) ) --> CC
 
Theoremgamcl 23967 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( _G `  A )  e. 
 CC )
 
Theoremeflgam 23968 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( exp `  ( log _G `  A ) )  =  ( _G `  A ) )
 
Theoremgamne0 23969 The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( _G `  A )  =/=  0 )
 
Theoremigamval 23970 Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  CC  ->  (1/ _G `  A )  =  if ( A  e.  ( ZZ  \  NN ) ,  0 ,  ( 1  /  ( _G `  A ) ) ) )
 
Theoremigamz 23971 Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  ( ZZ  \  NN )  ->  (1/ _G `  A )  =  0 )
 
Theoremigamgam 23972 Value of the inverse Gamma function in terms of the Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  (1/ _G
 `  A )  =  ( 1  /  ( _G `  A ) ) )
 
Theoremigamlgam 23973 Value of the inverse Gamma function in terms of the log-Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  (1/ _G
 `  A )  =  ( exp `  -u ( log _G `  A ) ) )
 
Theoremigamf 23974 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |- 1/ _G : CC --> CC
 
Theoremigamcl 23975 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  CC  ->  (1/ _G `  A )  e.  CC )
 
Theoremgamigam 23976 The Gamma function is the inverse of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( _G `  A )  =  ( 1  /  (1/ _G
 `  A ) ) )
 
Theoremlgamcvg 23977* The series  G converges to  log _G ( A )  +  log ( A ). (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( A  /  m )  +  1 ) ) ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN )
 ) )   =>    |-  ( ph  ->  seq 1
 (  +  ,  G ) 
 ~~>  ( ( log _G `  A )  +  ( log `  A ) ) )
 
Theoremlgamcvg2 23978* The series  G converges to  log _G ( A  +  1 ). (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( A  /  m )  +  1 ) ) ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN )
 ) )   =>    |-  ( ph  ->  seq 1
 (  +  ,  G ) 
 ~~>  ( log _G `  ( A  +  1 )
 ) )
 
Theoremgamcvg 23979* The pointwise exponential of the series  G converges to  _G ( A )  x.  A. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( A  /  m )  +  1 ) ) ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN )
 ) )   =>    |-  ( ph  ->  ( exp  o.  seq 1 (  +  ,  G ) )  ~~>  ( ( _G `  A )  x.  A ) )
 
Theoremlgamp1 23980 The functional equation of the (log) Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( log _G `  ( A  +  1 ) )  =  ( ( log _G `  A )  +  ( log `  A )
 ) )
 
Theoremgamp1 23981 The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( _G `  ( A  +  1 ) )  =  ( ( _G `  A )  x.  A ) )
 
Theoremgamcvg2lem 23982* Lemma for gamcvg2 23983. (Contributed by Mario Carneiro, 10-Jul-2017.)
 |-  F  =  ( m  e.  NN  |->  ( ( ( ( m  +  1 )  /  m )  ^c  A ) 
 /  ( ( A 
 /  m )  +  1 ) ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( A 
 /  m )  +  1 ) ) ) )   =>    |-  ( ph  ->  ( exp  o.  seq 1 (  +  ,  G ) )  =  seq 1
 (  x.  ,  F ) )
 
Theoremgamcvg2 23983* An infinite product expression for the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  F  =  ( m  e.  NN  |->  ( ( ( ( m  +  1 )  /  m )  ^c  A ) 
 /  ( ( A 
 /  m )  +  1 ) ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  seq 1
 (  x.  ,  F ) 
 ~~>  ( ( _G `  A )  x.  A ) )
 
Theoremregamcl 23984 The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  ( RR  \  ( ZZ  \  NN ) )  ->  ( _G `  A )  e. 
 RR )
 
Theoremrelgamcl 23985 The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  RR+  ->  ( log _G `  A )  e.  RR )
 
Theoremrpgamcl 23986 The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  RR+  ->  ( _G `  A )  e.  RR+ )
 
Theoremlgam1 23987 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( log _G `  1
 )  =  0
 
Theoremgam1 23988 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( _G `  1
 )  =  1
 
Theoremfacgam 23989 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  =  ( _G `  ( N  +  1 )
 ) )
 
Theoremgamfac 23990 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( N  e.  NN  ->  ( _G `  N )  =  ( ! `  ( N  -  1
 ) ) )
 
14.4  Basic number theory
 
14.4.1  Wilson's theorem
 
Theoremwilthlem1 23991 The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in  ZZ 
/  P ZZ are  1 and  -u 1  ==  P  -  1. (Note that from prmdiveq 14733,  ( N ^ ( P  - 
2 ) )  mod 
P is the modular inverse of  N in  ZZ  /  P ZZ. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( ( P  e.  Prime  /\  N  e.  (
 1 ... ( P  -  1 ) ) ) 
 ->  ( N  =  ( ( N ^ ( P  -  2 ) ) 
 mod  P )  <->  ( N  =  1  \/  N  =  ( P  -  1 ) ) ) )
 
Theoremwilthlem2 23992* Lemma for wilth 23994: induction step. The "hand proof" version of this theorem works by writing out the list of all numbers from  1 to  P  -  1 in pairs such that a number is paired with its inverse. Every number has a unique inverse different from itself except  1 and  P  -  1, and so each pair multiplies to  1, and  1 and  P  -  1  ==  -u 1 multiply to  -u 1, so the full product is equal to  -u 1. Here we make this precise by doing the product pair by pair.

The induction hypothesis says that every subset  S of  1 ... ( P  -  1 ) that is closed under inverse (i.e. all pairs are matched up) and contains 
P  -  1 multiplies to  -u 1  mod  P. Given such a set, we take out one element  z  =/=  P  -  1. If there are no such elements, then 
S  =  { P  -  1 } which forms the base case. Otherwise,  S  \  { z ,  z ^ -u 1 } is also closed under inverse and contains  P  -  1, so the induction hypothesis says that this equals  -u 1; and the remaining two elements are either equal to each other, in which case wilthlem1 23991 gives that  z  =  1 or  P  -  1, and we've already excluded the second case, so the product gives  1; or  z  =/=  z ^ -u 1 and their product is  1. In either case the accumulated product is unaffected. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)

 |-  T  =  (mulGrp ` fld )   &    |-  A  =  { x  e.  ~P (
 1 ... ( P  -  1 ) )  |  ( ( P  -  1 )  e.  x  /\  A. y  e.  x  ( ( y ^
 ( P  -  2
 ) )  mod  P )  e.  x ) }   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  S  e.  A )   &    |-  ( ph  ->  A. s  e.  A  ( s  C.  S  ->  ( ( T  gsumg  (  _I  |`  s ) )  mod  P )  =  ( -u 1  mod  P ) ) )   =>    |-  ( ph  ->  ( ( T  gsumg  (  _I  |`  S ) )  mod  P )  =  ( -u 1  mod  P ) )
 
Theoremwilthlem3 23993* Lemma for wilth 23994. Here we round out the argument of wilthlem2 23992 with the final step of the induction. The induction argument shows that every subset of  1 ... ( P  -  1 ) that is closed under inverse and contains  P  -  1 multiplies to  -u 1  mod  P, and clearly  1 ... ( P  -  1 ) itself is such a set. Thus, the product of all the elements is  -u 1, and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)
 |-  T  =  (mulGrp ` fld )   &    |-  A  =  { x  e.  ~P (
 1 ... ( P  -  1 ) )  |  ( ( P  -  1 )  e.  x  /\  A. y  e.  x  ( ( y ^
 ( P  -  2
 ) )  mod  P )  e.  x ) }   =>    |-  ( P  e.  Prime  ->  P  ||  ( ( ! `
  ( P  -  1 ) )  +  1 ) )
 
Theoremwilth 23994 Wilson's theorem. A number is prime iff it is greater or equal to  2 and  ( N  - 
1 ) ! is congruent to  -u 1,  mod  N, or alternatively if  N divides  ( N  - 
1 ) !  + 
1. In this part of the proof we show the relatively simple reverse implication; see wilthlem3 23993 for the forward implication. This is Metamath 100 proof #51. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  ( N  e.  Prime  <->  ( N  e.  ( ZZ>= `  2 )  /\  N  ||  ( ( ! `  ( N  -  1
 ) )  +  1 ) ) )
 
14.4.2  The Fundamental Theorem of Algebra
 
Theoremftalem1 23995* Lemma for fta 24004: "growth lemma". There exists some  r such that  F is arbitrarily close in proportion to its dominant term. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  T  =  ( sum_ k  e.  ( 0 ... ( N  -  1
 ) ) ( abs `  ( A `  k
 ) )  /  E )   =>    |-  ( ph  ->  E. r  e.  RR  A. x  e. 
 CC  ( r  < 
 ( abs `  x )  ->  ( abs `  (
 ( F `  x )  -  ( ( A `
  N )  x.  ( x ^ N ) ) ) )  <  ( E  x.  ( ( abs `  x ) ^ N ) ) ) )
 
Theoremftalem2 23996* Lemma for fta 24004. There exists some  r such that  F has magnitude greater than  F ( 0 ) outside the closed ball B(0,r). (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  U  =  if ( if (
 1  <_  s ,  s ,  1 )  <_  T ,  T ,  if ( 1  <_  s ,  s ,  1 ) )   &    |-  T  =  ( ( abs `  ( F `  0 ) ) 
 /  ( ( abs `  ( A `  N ) )  /  2
 ) )   =>    |-  ( ph  ->  E. r  e.  RR+  A. x  e.  CC  ( r  <  ( abs `  x )  ->  ( abs `  ( F `  0 ) )  < 
 ( abs `  ( F `  x ) ) ) )
 
Theoremftalem3 23997* Lemma for fta 24004. There exists a global minimum of the function  abs  o.  F. The proof uses a circle of radius  r where  r is the value coming from ftalem1 23995; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  D  =  { y  e.  CC  |  ( abs `  y
 )  <_  R }   &    |-  J  =  ( TopOpen ` fld )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  CC  ( R  <  ( abs `  x )  ->  ( abs `  ( F `  0 ) )  <  ( abs `  ( F `  x ) ) ) )   =>    |-  ( ph  ->  E. z  e.  CC  A. x  e. 
 CC  ( abs `  ( F `  z ) ) 
 <_  ( abs `  ( F `  x ) ) )
 
Theoremftalem4 23998* Lemma for fta 24004: Closure of the auxiliary variables for ftalem5 23999. (Contributed by Mario Carneiro, 20-Sep-2014.) (Revised by AV, 28-Sep-2020.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( F `  0 )  =/=  0
 )   &    |-  K  = inf ( { n  e.  NN  |  ( A `  n )  =/=  0 } ,  RR ,  <  )   &    |-  T  =  ( -u ( ( F `
  0 )  /  ( A `  K ) )  ^c  ( 1  /  K ) )   &    |-  U  =  ( ( abs `  ( F `  0 ) ) 
 /  ( sum_ k  e.  ( ( K  +  1 ) ... N ) ( abs `  (
 ( A `  k
 )  x.  ( T ^ k ) ) )  +  1 ) )   &    |-  X  =  if ( 1  <_  U ,  1 ,  U )   =>    |-  ( ph  ->  (
 ( K  e.  NN  /\  ( A `  K )  =/=  0 )  /\  ( T  e.  CC  /\  U  e.  RR+  /\  X  e.  RR+ ) ) )
 
Theoremftalem5 23999* Lemma for fta 24004: Main proof. We have already shifted the minimum found in ftalem3 23997 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let  K be the lowest term in the polynomial that is nonzero, and let  T be a  K-th root of  -u F ( 0 )  /  A
( K ). Then an evaluation of  F ( T X ) where  X is a sufficiently small positive number yields  F ( 0 ) for the first term and 
-u F ( 0 )  x.  X ^ K for the  K-th term, and all higher terms are bounded because  X is small. Thus,  abs ( F ( T X ) )  <_  abs ( F ( 0 ) ) ( 1  -  X ^ K )  <  abs ( F ( 0 ) ), in contradiction to our choice of  F ( 0 ) as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.) (Revised by AV, 28-Sep-2020.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( F `  0 )  =/=  0
 )   &    |-  K  = inf ( { n  e.  NN  |  ( A `  n )  =/=  0 } ,  RR ,  <  )   &    |-  T  =  ( -u ( ( F `
  0 )  /  ( A `  K ) )  ^c  ( 1  /  K ) )   &    |-  U  =  ( ( abs `  ( F `  0 ) ) 
 /  ( sum_ k  e.  ( ( K  +  1 ) ... N ) ( abs `  (
 ( A `  k
 )  x.  ( T ^ k ) ) )  +  1 ) )   &    |-  X  =  if ( 1  <_  U ,  1 ,  U )   =>    |-  ( ph  ->  E. x  e.  CC  ( abs `  ( F `  x ) )  <  ( abs `  ( F `  0 ) ) )
 
Theoremftalem4OLD 24000* Lemma for fta 24004: Closure of the auxiliary variables for ftalem5 23999. (Contributed by Mario Carneiro, 20-Sep-2014.) Obsolete version of ftalem4 23998 as of 1-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( F `  0 )  =/=  0
 )   &    |-  K  =  sup ( { n  e.  NN  |  ( A `  n )  =/=  0 } ,  RR ,  `'  <  )   &    |-  T  =  ( -u ( ( F `
  0 )  /  ( A `  K ) )  ^c  ( 1  /  K ) )   &    |-  U  =  ( ( abs `  ( F `  0 ) ) 
 /  ( sum_ k  e.  ( ( K  +  1 ) ... N ) ( abs `  (
 ( A `  k
 )  x.  ( T ^ k ) ) )  +  1 ) )   &    |-  X  =  if ( 1  <_  U ,  1 ,  U )   =>    |-  ( ph  ->  (
 ( K  e.  NN  /\  ( A `  K )  =/=  0 )  /\  ( T  e.  CC  /\  U  e.  RR+  /\  X  e.  RR+ ) ) )
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