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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremleibpisum 20201 The Leibniz formula for  pi. This version of leibpi 20200 looks nicer but does not assert that the series is convergent so is not as practically useful. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |- 
 sum_ n  e.  NN0  (
 ( -u 1 ^ n )  /  ( ( 2  x.  n )  +  1 ) )  =  ( pi  /  4
 )
 
Theoremlog2cnv 20202 Using the Taylor series for arctan ( _i  /  3
), produce a rapidly convergent series for  log 2. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  F  =  ( n  e.  NN0  |->  ( 2 
 /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) )   =>    |-  seq  0 (  +  ,  F )  ~~>  ( log `  2
 )
 
Theoremlog2tlbnd 20203* Bound the error term in the series of log2cnv 20202. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( N  e.  NN0  ->  ( ( log `  2
 )  -  sum_ n  e.  ( 0 ... ( N  -  1 ) ) ( 2  /  (
 ( 3  x.  (
 ( 2  x.  n )  +  1 )
 )  x.  ( 9 ^ n ) ) ) )  e.  (
 0 [,] ( 3  /  ( ( 4  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
 9 ^ N ) ) ) ) )
 
13.3.8  The Birthday Problem
 
Theoremlog2ublem1 20204 Lemma for log2ub 20207. The proof of log2ub 20207, which is simply the evaluation of log2tlbnd 20203 for  N  =  4, takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator  d (usually a large power of  10) and work with closest approximations of the form  n  /  d for some integer  n instead. It turns out that for our purposes it is sufficient to take  d  =  ( 3 ^ 7 )  x.  5  x.  7, which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7
 ) )  x.  A )  <_  B   &    |-  A  e.  RR   &    |-  D  e.  NN0   &    |-  E  e.  NN   &    |-  B  e.  NN0   &    |-  F  e.  NN0   &    |-  C  =  ( A  +  ( D  /  E ) )   &    |-  ( B  +  F )  =  G   &    |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  <_  ( E  x.  F )   =>    |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  <_  G
 
Theoremlog2ublem2 20205* Lemma for log2ub 20207. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7
 ) )  x.  sum_ n  e.  ( 0 ...
 K ) ( 2 
 /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) ) 
 <_  ( 2  x.  B )   &    |-  B  e.  NN0   &    |-  F  e.  NN0   &    |-  N  e.  NN0   &    |-  ( N  -  1
 )  =  K   &    |-  ( B  +  F )  =  G   &    |-  M  e.  NN0   &    |-  ( M  +  N )  =  3   &    |-  ( ( 5  x.  7 )  x.  ( 9 ^ M ) )  =  (
 ( ( 2  x.  N )  +  1 )  x.  F )   =>    |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7
 ) )  x.  sum_ n  e.  ( 0 ...
 N ) ( 2 
 /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) ) 
 <_  ( 2  x.  G )
 
Theoremlog2ublem3 20206 Lemma for log2ub 20207. In decimal, this is a proof that the first four terms of the series for 
log 2 is less than  5 3
0 5 6  / 
7 6 5 4 5. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7
 ) )  x.  sum_ n  e.  ( 0 ... 3 ) ( 2 
 /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) ) 
 <_ ;;;; 5 3 0 5 6
 
Theoremlog2ub 20207  log 2 is less than  2 5 3  /  3
6 5. If written in decimal, this is because  log 2  = 0.693147... is less than 253/365 = 0.693151... , so this is a very tight bound, at five decimal places. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( log `  2
 )  <  (;; 2 5 3  / ;; 3 6 5 )
 
Theorembirthdaylem1 20208* Lemma for birthday 20211. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  S  =  { f  |  f : ( 1
 ... K ) --> ( 1
 ... N ) }   &    |-  T  =  { f  |  f : ( 1 ...
 K ) -1-1-> ( 1
 ... N ) }   =>    |-  ( T  C_  S  /\  S  e.  Fin  /\  ( N  e.  NN  ->  S  =/=  (/) ) )
 
Theorembirthdaylem2 20209* For general  N and  K, count the fraction of injective functions from  1 ... K to  1 ... N. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  S  =  { f  |  f : ( 1
 ... K ) --> ( 1
 ... N ) }   &    |-  T  =  { f  |  f : ( 1 ...
 K ) -1-1-> ( 1
 ... N ) }   =>    |-  (
 ( N  e.  NN  /\  K  e.  ( 0
 ... N ) ) 
 ->  ( ( # `  T )  /  ( # `  S ) )  =  ( exp `  sum_ k  e.  (
 0 ... ( K  -  1 ) ) ( log `  ( 1  -  ( k  /  N ) ) ) ) )
 
Theorembirthdaylem3 20210* For general  N and  K, upper-bound the fraction of injective functions from  1 ... K to  1 ... N. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  S  =  { f  |  f : ( 1
 ... K ) --> ( 1
 ... N ) }   &    |-  T  =  { f  |  f : ( 1 ...
 K ) -1-1-> ( 1
 ... N ) }   =>    |-  (
 ( K  e.  NN0  /\  N  e.  NN )  ->  ( ( # `  T )  /  ( # `  S ) )  <_  ( exp `  -u ( ( ( ( K ^ 2
 )  -  K ) 
 /  2 )  /  N ) ) )
 
Theorembirthday 20211* The Birthday Problem. There is a more than even chance that out of 23 people in a room, at least two of them have the same birthday. Mathematically, this is asserting that for  K  =  2 3 and  N  =  3 6 5, fewer than half of the set of all functions from  1 ... K to  1 ... N are injective. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  S  =  { f  |  f : ( 1
 ... K ) --> ( 1
 ... N ) }   &    |-  T  =  { f  |  f : ( 1 ...
 K ) -1-1-> ( 1
 ... N ) }   &    |-  K  = ; 2 3   &    |-  N  = ;; 3 6 5   =>    |-  ( ( # `  T )  /  ( # `  S ) )  <  ( 1 
 /  2 )
 
13.3.9  Areas in R^2
 
Syntaxcarea 20212 Area of regions in the complex plane.
 class area
 
Definitiondf-area 20213* Define the area of a subset of  RR  X.  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |- area  =  ( s  e.  {
 t  e.  ~P ( RR  X.  RR )  |  ( A. x  e. 
 RR  ( t " { x } )  e.  ( `' vol " RR )  /\  ( x  e. 
 RR  |->  ( vol `  (
 t " { x }
 ) ) )  e.  L ^1 ) }  |->  S. RR ( vol `  ( s " { x } ) )  _d x )
 
Theoremdmarea 20214* The domain of the area function is the set of finitely measurable subsets of  RR  X.  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( A  e.  dom area  <->  ( A  C_  ( RR  X.  RR )  /\  A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  ( x  e. 
 RR  |->  ( vol `  ( A " { x }
 ) ) )  e.  L ^1 ) )
 
Theoremareambl 20215 The fibers of a measurable region are finitely meaurable subsets of  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( S  e.  dom area  /\  A  e.  RR )  ->  ( ( S " { A } )  e. 
 dom  vol  /\  ( vol `  ( S " { A } ) )  e. 
 RR ) )
 
Theoremareass 20216 A measurable region is a subset of 
RR  X.  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( S  e.  dom area  ->  S  C_  ( RR  X.  RR ) )
 
Theoremdfarea 20217* Rewrite df-area 20213 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  (
 s " { x }
 ) )  _d x )
 
Theoremareaf 20218 Area meaurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |- area : dom area
 --> ( 0 [,)  +oo )
 
Theoremareacl 20219 The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( S  e.  dom area  ->  (area `  S )  e. 
 RR )
 
Theoremareage0 20220 The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( S  e.  dom area  -> 
 0  <_  (area `  S ) )
 
Theoremareaval 20221* The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( S  e.  dom area  ->  (area `  S )  =  S. RR ( vol `  ( S " { x } ) )  _d x )
 
13.3.10  More miscellaneous converging sequences
 
Theoremrlimcnp 20222* Relate a limit of a real-valued sequence at infinity to the continuity of the function  S ( y )  =  R ( 1  /  y ) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  ( ph  ->  A  C_  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  0  e.  A )   &    |-  ( ph  ->  B  C_  RR+ )   &    |-  (
 ( ph  /\  x  e.  A )  ->  R  e.  CC )   &    |-  ( ( ph  /\  x  e.  RR+ )  ->  ( x  e.  A  <->  ( 1  /  x )  e.  B ) )   &    |-  ( x  =  0  ->  R  =  C )   &    |-  ( x  =  (
 1  /  y )  ->  R  =  S )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  A )   =>    |-  ( ph  ->  (
 ( y  e.  B  |->  S )  ~~> r  C  <->  ( x  e.  A  |->  R )  e.  ( ( K  CnP  J ) `  0 ) ) )
 
Theoremrlimcnp2 20223* Relate a limit of a real-valued sequence at infinity to the continuity of the function  S ( y )  =  R ( 1  /  y ) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  ( ph  ->  A  C_  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  0  e.  A )   &    |-  ( ph  ->  B  C_  RR )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  S  e.  CC )   &    |-  (
 ( ph  /\  y  e.  RR+ )  ->  ( y  e.  B  <->  ( 1  /  y )  e.  A ) )   &    |-  ( y  =  ( 1  /  x )  ->  S  =  R )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  A )   =>    |-  ( ph  ->  (
 ( y  e.  B  |->  S )  ~~> r  C  <->  ( x  e.  A  |->  if ( x  =  0 ,  C ,  R ) )  e.  ( ( K  CnP  J ) `  0 ) ) )
 
Theoremrlimcnp3 20224* Relate a limit of a real-valued sequence at infinity to the continuity of the function  S ( y )  =  R ( 1  /  y ) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  y  e.  RR+ )  ->  S  e.  CC )   &    |-  (
 y  =  ( 1 
 /  x )  ->  S  =  R )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  (
 ( y  e.  RR+  |->  S )  ~~> r  C  <->  ( x  e.  ( 0 [,)  +oo )  |->  if ( x  =  0 ,  C ,  R ) )  e.  ( ( K  CnP  J ) `  0 ) ) )
 
Theoremxrlimcnp 20225* Relate a limit of a real-valued sequence at infinity to the continuity of the corresponding extended real function at  +oo. Since any  ~~> r limit can be written in the form on the left side of the implication, this shows that real limits are a special case of topological continuity at a point. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  ( ph  ->  A  =  ( B  u.  {  +oo } ) )   &    |-  ( ph  ->  B  C_  RR )   &    |-  ( ( ph  /\  x  e.  A )  ->  R  e.  CC )   &    |-  ( x  = 
 +oo  ->  R  =  C )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( (ordTop `  <_  )t  A )   =>    |-  ( ph  ->  (
 ( x  e.  B  |->  R )  ~~> r  C  <->  ( x  e.  A  |->  R )  e.  ( ( K  CnP  J ) `  +oo )
 ) )
 
Theoremefrlim 20226* The limit of the sequence  ( 1  +  A  /  k ) ^
k is the exponential function. This is often taken as an alternate definition of the exponential function (see also dfef2 20227). (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 0 ( ball `  ( abs  o. 
 -  ) ) ( 1  /  ( ( abs `  A )  +  1 ) ) )   =>    |-  ( A  e.  CC  ->  ( k  e.  RR+  |->  ( ( 1  +  ( A  /  k
 ) )  ^ c  k ) )  ~~> r  ( exp `  A )
 )
 
Theoremdfef2 20227* The limit of the sequence  ( 1  +  A  /  k ) ^
k as  k goes to  +oo is  ( exp `  A
). This is another common definition of  _e. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  A  e.  CC )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  k )  =  ( ( 1  +  ( A  /  k
 ) ) ^ k
 ) )   =>    |-  ( ph  ->  F  ~~>  ( exp `  A )
 )
 
Theoremcxplim 20228* A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)
 |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( 1 
 /  ( n  ^ c  A ) ) )  ~~> r  0 )
 
Theoremsqrlim 20229 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( n  e.  RR+  |->  ( 1  /  ( sqr `  n ) ) )  ~~> r  0
 
Theoremrlimcxp 20230* Any power to a positive exponent of a converging sequence also converges. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( ( ph  /\  n  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( n  e.  A  |->  B )  ~~> r  0 )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( n  e.  A  |->  ( B 
 ^ c  C ) )  ~~> r  0 )
 
Theoremo1cxp 20231* An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  0 
 <_  ( Re `  C ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B 
 ^ c  C ) )  e.  O ( 1 ) )
 
Theoremcxp2limlem 20232* A linear factor grows slower than any exponential with base greater than  1. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  RR  /\  1  <  A )  ->  ( n  e.  RR+  |->  ( n  /  ( A  ^ c  n ) ) )  ~~> r  0 )
 
Theoremcxp2lim 20233* Any power grows slower than any exponential with base greater than  1. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B ) 
 ->  ( n  e.  RR+  |->  ( ( n  ^ c  A )  /  ( B  ^ c  n ) ) )  ~~> r  0 )
 
Theoremcxploglim 20234* 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 20235* 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 )
 
Theoremdivsqrsumlem 20236* Lemma for divsqrsum 20238 and divsqrsum2 20239. (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 20237* The function  F used in divsqrsum 20238 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 20238* 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
 
Theoremdivsqrsum2 20239* 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 ) ) )
 
Theoremdivsqrsumo1 20240* 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 ) )
 
13.3.11  Inequality of arithmetic and geometric means
 
Theoremcvxcl 20241* 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 20242* 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 20243* Lemma for jensen 20245. (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 20244* Lemma for jensen 20245. (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  o F  x.  X )  |`  B ) )  /  S )  e.  D )   &    |-  ( ph  ->  ( F `  ( (fld  gsumg  ( ( T  o F  x.  X )  |`  B ) )  /  S ) )  <_  ( (fld 
 gsumg  ( ( T  o F  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )   =>    |-  ( ph  ->  ( ( (fld 
 gsumg  ( ( T  o F  x.  X )  |`  ( B  u.  { z } ) ) ) 
 /  L )  e.  D  /\  ( F `
  ( (fld  gsumg  ( ( T  o F  x.  X )  |`  ( B  u.  { z } ) ) ) 
 /  L ) ) 
 <_  ( (fld 
 gsumg  ( ( T  o F  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) ) 
 /  L ) ) )
 
Theoremjensen 20245* Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (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  ->  ( (
 (fld  gsumg  ( T  o F  x.  X ) )  /  (fld  gsumg  T ) )  e.  D  /\  ( F `  (
 (fld  gsumg  ( T  o F  x.  X ) )  /  (fld  gsumg  T ) ) )  <_  ( (fld 
 gsumg  ( T  o F  x.  ( F  o.  X ) ) )  /  (fld  gsumg  T ) ) ) )
 
Theoremamgmlem 20246 Lemma for amgm 20247. (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 20247 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). (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 ) ) )
 
13.3.12  Euler-Mascheroni constant
 
Syntaxcem 20248 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)
 class  gamma
 
Definitiondf-em 20249 Define the Euler-Macheroni 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 20258. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  =  sum_ k  e. 
 NN  ( ( 1 
 /  k )  -  ( log `  ( 1  +  ( 1  /  k
 ) ) ) )
 
Theoremlogdifbnd 20250 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)
 |-  ( A  e.  RR+  ->  ( ( log `  ( A  +  1 )
 )  -  ( log `  A ) )  <_  ( 1  /  A ) )
 
Theorememcllem1 20251* Lemma for emcl 20258. 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 20252* Lemma for emcl 20258. 
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 20253* Lemma for emcl 20258. 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 20254* Lemma for emcl 20258. 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 20255* Lemma for emcl 20258. The partial sums of the series  T, which is used in the definition df-em 20249, 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 20256* Lemma for emcl 20258. 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 20257* Lemma for emcl 20258 and harmonicbnd 20259. 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 20258 Closure and bounds for the Euler-Macheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  e.  ( ( 1  -  ( log `  2
 ) ) [,] 1
 )
 
Theoremharmonicbnd 20259* 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 20260* 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 20261 The Euler-Macheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  e.  RR
 
Theorememgt0 20262 The Euler-Macheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  0  <  gamma
 
Theoremharmonicbnd3 20263* 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 20264* 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 20265* 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 20266* 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 20267* 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 ) ) ) )
 
13.4  Basic number theory
 
13.4.1  Wilson's theorem
 
Theoremwilthlem1 20268 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 12816,  ( 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 20269* Lemma for wilth 20271: 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 20268 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.)

 |-  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 20270* Lemma for wilth 20271. Here we round out the argument of wilthlem2 20269 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.)
 |-  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 20271 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 20270 for the forward implication. (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 ) ) )
 
13.4.2  The Fundamental Theorem of Algebra
 
Theoremftalem1 20272* Lemma for fta 20279: "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 20273* Lemma for fta 20279. 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 20274* Lemma for fta 20279. 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 20272; 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 20275* Lemma for fta 20279: Closure of the auxiliary variables for ftalem5 20276. (Contributed by Mario Carneiro, 20-Sep-2014.)
 |-  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+ ) ) )
 
Theoremftalem5 20276* Lemma for fta 20279: Main proof. We have already shifted the minimum found in ftalem3 20274 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.)
 |-  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  ->  E. x  e.  CC  ( abs `  ( F `  x ) )  <  ( abs `  ( F `  0 ) ) )
 
Theoremftalem6 20277* Lemma for fta 20279: Discharge the auxiliary variables in ftalem5 20276. (Contributed by Mario Carneiro, 20-Sep-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( F `  0 )  =/=  0
 )   =>    |-  ( ph  ->  E. x  e.  CC  ( abs `  ( F `  x ) )  <  ( abs `  ( F `  0 ) ) )
 
Theoremftalem7 20278* Lemma for fta 20279. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  A  =  (coeff `  F )   &    |-  N  =  (deg `  F )   &    |-  ( ph  ->  F  e.  (Poly `  S ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( F `  X )  =/=  0 )   =>    |-  ( ph  ->  -.  A. x  e.  CC  ( abs `  ( F `  X ) )  <_  ( abs `  ( F `  x ) ) )
 
Theoremfta 20279* The Fundamental Theorem of Algebra. Any polynomial with positive degree (i.e. non-constant) has a root. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  e. 
 NN )  ->  E. z  e.  CC  ( F `  z )  =  0
 )
 
13.4.3  The Basel problem (ζ(2) = π2/6)
 
Theorembasellem1 20280 Lemma for basel 20289. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.)
 |-  N  =  ( ( 2  x.  M )  +  1 )   =>    |-  ( ( M  e.  NN  /\  K  e.  ( 1 ... M ) )  ->  ( ( K  x.  pi ) 
 /  N )  e.  ( 0 (,) ( pi  /  2 ) ) )
 
Theorembasellem2 20281* Lemma for basel 20289. Show that  P is a polynomial of degree  M, and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)
 |-  N  =  ( ( 2  x.  M )  +  1 )   &    |-  P  =  ( t  e.  CC  |->  sum_
 j  e.  ( 0
 ... M ) ( ( ( N  _C  ( 2  x.  j
 ) )  x.  ( -u 1 ^ ( M  -  j ) ) )  x.  ( t ^ j ) ) )   =>    |-  ( M  e.  NN  ->  ( P  e.  (Poly `  CC )  /\  (deg `  P )  =  M  /\  (coeff `  P )  =  ( n  e.  NN0  |->  ( ( N  _C  ( 2  x.  n ) )  x.  ( -u 1 ^ ( M  -  n ) ) ) ) ) )
 
Theorembasellem3 20282* Lemma for basel 20289. Using the binomial theorem and de Moivre's formula, we have the identity  _e ^ _i N x  /  ( sin x
) ^ n  =  sum_ m  e.  ( 0 ... N
) ( N  _C  m ) ( _i
^ m ) ( cot x ) ^
( N  -  m
), so taking imaginary parts yields  sin ( N x )  /  ( sin x
) ^ N  =  sum_ j  e.  ( 0 ... M
) ( N  _C  2 j ) (
-u 1 ) ^
( M  -  j
)  ( cot x
) ^ ( -u
2 j )  =  P ( ( cot x ) ^ 2 ), where  N  =  2 M  +  1. (Contributed by Mario Carneiro, 30-Jul-2014.)
 |-  N  =  ( ( 2  x.  M )  +  1 )   &    |-  P  =  ( t  e.  CC  |->  sum_
 j  e.  ( 0
 ... M ) ( ( ( N  _C  ( 2  x.  j
 ) )  x.  ( -u 1 ^ ( M  -  j ) ) )  x.  ( t ^ j ) ) )   =>    |-  ( ( M  e.  NN  /\  A  e.  (
 0 (,) ( pi  / 
 2 ) ) ) 
 ->  ( P `  (
 ( tan `  A ) ^ -u 2 ) )  =  ( ( sin `  ( N  x.  A ) )  /  (
 ( sin `  A ) ^ N ) ) )
 
Theorembasellem4 20283* Lemma for basel 20289. By basellem3 20282, the expression  P ( ( cot x ) ^
2 )  =  sin ( N x )  / 
( sin x ) ^ N goes to zero whenever  x  =  n pi  /  N for some  n  e.  ( 1 ... M
), so this function enumerates  M distinct roots of a degree-  M polynomial, which must therefore be all the roots by fta1 19650. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  N  =  ( ( 2  x.  M )  +  1 )   &    |-  P  =  ( t  e.  CC  |->  sum_
 j  e.  ( 0
 ... M ) ( ( ( N  _C  ( 2  x.  j
 ) )  x.  ( -u 1 ^ ( M  -  j ) ) )  x.  ( t ^ j ) ) )   &    |-  T  =  ( n  e.  ( 1
 ... M )  |->  ( ( tan `  (
 ( n  x.  pi )  /  N ) ) ^ -u 2 ) )   =>    |-  ( M  e.  NN  ->  T : ( 1
 ... M ) -1-1-onto-> ( `' P " { 0 } ) )
 
Theorembasellem5 20284* Lemma for basel 20289. Using vieta1 19654, we can calculate the sum of the roots of  P as the quotient of the top two coefficients, and since the function  T enumerates the roots, we are left with an equation that sums the  cot ^ 2 function at the  M different roots. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  N  =  ( ( 2  x.  M )  +  1 )   &    |-  P  =  ( t  e.  CC  |->  sum_
 j  e.  ( 0
 ... M ) ( ( ( N  _C  ( 2  x.  j
 ) )  x.  ( -u 1 ^ ( M  -  j ) ) )  x.  ( t ^ j ) ) )   &    |-  T  =  ( n  e.  ( 1
 ... M )  |->  ( ( tan `  (
 ( n  x.  pi )  /  N ) ) ^ -u 2 ) )   =>    |-  ( M  e.  NN  -> 
 sum_ k  e.  (
 1 ... M ) ( ( tan `  (
 ( k  x.  pi )  /  N ) ) ^ -u 2 )  =  ( ( ( 2  x.  M )  x.  ( ( 2  x.  M )  -  1
 ) )  /  6
 ) )
 
Theorembasellem6 20285 Lemma for basel 20289. The function  G goes to zero because it is bounded by  1  /  n. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  G  =  ( n  e.  NN  |->  ( 1 
 /  ( ( 2  x.  n )  +  1 ) ) )   =>    |-  G 
 ~~>  0
 
Theorembasellem7 20286 Lemma for basel 20289. The function  1  +  A  x.  G for any fixed  A goes to  1. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  G  =  ( n  e.  NN  |->  ( 1 
 /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  A  e.  CC   =>    |-  ( ( NN  X.  { 1 } )  o F  +  ( ( NN  X.  { A } )  o F  x.  G ) )  ~~>  1
 
Theorembasellem8 20287* Lemma for basel 20289. The function  F of partial sums of the inverse squares is bounded below by  J and above by  K, obtained by summing the inequality 
cot ^ 2 x  <_ 
1  /  x ^
2  <_  csc ^ 2 x  =  cot ^
2 x  +  1 over the  M roots of the polynomial  P, and applying the identity basellem5 20284. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  G  =  ( n  e.  NN  |->  ( 1 
 /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  F  =  seq  1 (  +  ,  ( n  e.  NN  |->  ( n ^ -u 2 ) ) )   &    |-  H  =  ( ( NN  X.  {
 ( ( pi ^
 2 )  /  6
 ) } )  o F  x.  ( ( NN  X.  { 1 } )  o F  -  G ) )   &    |-  J  =  ( H  o F  x.  ( ( NN  X.  { 1 } )  o F  +  ( ( NN  X.  { -u 2 } )  o F  x.  G ) ) )   &    |-  K  =  ( H  o F  x.  (
 ( NN  X.  {
 1 } )  o F  +  G ) )   &    |-  N  =  ( ( 2  x.  M )  +  1 )   =>    |-  ( M  e.  NN  ->  ( ( J `  M )  <_  ( F `  M )  /\  ( F `
  M )  <_  ( K `  M ) ) )
 
Theorembasellem9 20288* Lemma for basel 20289. Since by basellem8 20287 
F is bounded by two expressions that tend to  pi ^ 2  / 
6,  F must also go to  pi ^ 2  /  6 by the squeeze theorem climsqz 12079. But the series  F is exactly the partial sums of 
k ^ -u 2, so it follows that this is also the value of the infinite sum  sum_ k  e.  NN ( k ^ -u 2
). (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  G  =  ( n  e.  NN  |->  ( 1 
 /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  F  =  seq  1 (  +  ,  ( n  e.  NN  |->  ( n ^ -u 2 ) ) )   &    |-  H  =  ( ( NN  X.  {
 ( ( pi ^
 2 )  /  6
 ) } )  o F  x.  ( ( NN  X.  { 1 } )  o F  -  G ) )   &    |-  J  =  ( H  o F  x.  ( ( NN  X.  { 1 } )  o F  +  ( ( NN  X.  { -u 2 } )  o F  x.  G ) ) )   &    |-  K  =  ( H  o F  x.  (
 ( NN  X.  {
 1 } )  o F  +  G ) )   =>    |- 
 sum_ k  e.  NN  ( k ^ -u 2
 )  =  ( ( pi ^ 2 ) 
 /  6 )
 
Theorembasel 20289 The sum of the inverse squares is 
pi ^ 2  / 
6. This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). (Contributed by Mario Carneiro, 30-Jul-2014.)
 |- 
 sum_ k  e.  NN  ( k ^ -u 2
 )  =  ( ( pi ^ 2 ) 
 /  6 )
 
13.4.4  Number-theoretical functions
 
Syntaxccht 20290 Extend class notation with the first Chebyshev function.
 class  theta
 
Syntaxcvma 20291 Extend class notation with the von Mangoldt function.
 class Λ
 
Syntaxcchp 20292 Extend class notation with the second Chebyshev function.
 class ψ
 
Syntaxcppi 20293 Extend class notation with the prime Pi function.
 class π
 
Syntaxcmu 20294 Extend class notation with the Möbius function.
 class  mmu
 
Syntaxcsgm 20295 Extend class notation with the divisor function.
 class  sigma
 
Definitiondf-cht 20296* Define the first Chebyshev function, which adds up the logarithms of all primes less than  x. The symbol used to represent this function is sometimes the variant greek letter theta shown here and sometimes the greek letter psi, ψ; however, this notation can also refer to the second Chebyshev function, which adds up the logarithms of prime powers instead. See https://en.wikipedia.org/wiki/Chebyshev_function for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |- 
 theta  =  ( x  e.  RR  |->  sum_ p  e.  (
 ( 0 [,] x )  i^i  Prime ) ( log `  p ) )
 
Definitiondf-vma 20297* Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |- Λ  =  ( x  e.  NN  |->  [_
 { p  e.  Prime  |  p  ||  x }  /  s ]_ if (
 ( # `  s )  =  1 ,  ( log `  U. s ) ,  0 ) )
 
Definitiondf-chp 20298* Define the second Chebyshev function, which adds up the logarithms of the primes corresponding to the prime powers less than  x. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |- ψ  =  ( x  e.  RR  |->  sum_
 n  e.  ( 1
 ... ( |_ `  x ) ) (Λ `  n ) )
 
Definitiondf-ppi 20299 Define the prime π function, which counts the number of primes less than or equal to  x. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |- π  =  ( x  e.  RR  |->  ( # `  ( ( 0 [,] x )  i^i  Prime ) ) )
 
Definitiondf-mu 20300* Define the Möbius function, which is zero for non-squarefree numbers and is  -u 1 or  1 for squarefree numbers according as to the number of prime divisors of the number is even or odd. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |- 
 mmu  =  ( x  e.  NN  |->  if ( E. p  e.  Prime  ( p ^
 2 )  ||  x ,  0 ,  ( -u 1 ^ ( # ` 
 { p  e.  Prime  |  p  ||  x }
 ) ) ) )
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