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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremharmonicbnd3 20301* 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 20302* 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 20303* 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 20304* 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 20305* 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 20306 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 12854,  ( 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 20307* Lemma for wilth 20309: 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 20306 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 20308* Lemma for wilth 20309. Here we round out the argument of wilthlem2 20307 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 20309 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 20308 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 20310* Lemma for fta 20317: "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 20311* Lemma for fta 20317. 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 20312* Lemma for fta 20317. 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 20310; 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 20313* Lemma for fta 20317: Closure of the auxiliary variables for ftalem5 20314. (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 20314* Lemma for fta 20317: Main proof. We have already shifted the minimum found in ftalem3 20312 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 20315* Lemma for fta 20317: Discharge the auxiliary variables in ftalem5 20314. (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 20316* Lemma for fta 20317. 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 20317* 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 20318 Lemma for basel 20327. 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 20319* Lemma for basel 20327. 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 20320* Lemma for basel 20327. 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 20321* Lemma for basel 20327. By basellem3 20320, 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 19688. (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 20322* Lemma for basel 20327. Using vieta1 19692, 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 20323 Lemma for basel 20327. 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 20324 Lemma for basel 20327. 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 20325* Lemma for basel 20327. 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 20322. (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 20326* Lemma for basel 20327. Since by basellem8 20325 
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 12114. 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 20327 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 20328 Extend class notation with the first Chebyshev function.
 class  theta
 
Syntaxcvma 20329 Extend class notation with the von Mangoldt function.
 class Λ
 
Syntaxcchp 20330 Extend class notation with the second Chebyshev function.
 class ψ
 
Syntaxcppi 20331 Extend class notation with the prime Pi function.
 class π
 
Syntaxcmu 20332 Extend class notation with the Möbius function.
 class  mmu
 
Syntaxcsgm 20333 Extend class notation with the divisor function.
 class  sigma
 
Definitiondf-cht 20334* 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 20335* 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 20336* 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 20337 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 20338* 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 }
 ) ) ) )
 
Definitiondf-sgm 20339* Define the divisor function, which counts the number of divisors of  n, to the power  x. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |- 
 sigma  =  ( x  e.  CC ,  n  e. 
 NN  |->  sum_ k  e.  { p  e.  NN  |  p  ||  n }  ( k 
 ^ c  x ) )
 
Theoremefnnfsumcl 20340* Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  ( exp `  B )  e. 
 NN )   =>    |-  ( ph  ->  ( exp `  sum_ k  e.  A  B )  e.  NN )
 
Theoremppisval 20341 The set of primes less than  A expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR  ->  ( ( 0 [,]
 A )  i^i  Prime )  =  ( ( 2
 ... ( |_ `  A ) )  i^i  Prime )
 )
 
Theoremppisval2 20342 The set of primes less than  A expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  RR  /\  2  e.  ( ZZ>=
 `  M ) ) 
 ->  ( ( 0 [,]
 A )  i^i  Prime )  =  ( ( M
 ... ( |_ `  A ) )  i^i  Prime )
 )
 
Theoremppifi 20343 The set of primes less than  A is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  RR  ->  ( ( 0 [,]
 A )  i^i  Prime )  e.  Fin )
 
Theoremsgmss 20344* The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  NN  ->  { p  e.  NN  |  p  ||  A }  C_  ( 1 ... A ) )
 
Theoremprmdvdsfi 20345* The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  NN  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
 
Theoremchtf 20346 Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |- 
 theta : RR --> RR
 
Theoremchtcl 20347 Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
 
Theoremchtval 20348* Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  RR  ->  ( theta `  A )  =  sum_ p  e.  (
 ( 0 [,] A )  i^i  Prime ) ( log `  p ) )
 
Theoremefchtcl 20349 The Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  ( exp `  ( theta `  A ) )  e.  NN )
 
Theoremchtge0 20350 The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  RR  ->  0  <_  ( theta `  A ) )
 
Theoremvmaval 20351* Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  S  =  { p  e.  Prime  |  p  ||  A }   =>    |-  ( A  e.  NN  ->  (Λ `  A )  =  if ( ( # `  S )  =  1 ,  ( log `  U. S ) ,  0 )
 )
 
Theoremisppw 20352* Two ways to say that  A is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  NN  ->  ( (Λ `  A )  =/=  0  <->  E! p  e.  Prime  p 
 ||  A ) )
 
Theoremisppw2 20353* Two ways to say that  A is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  NN  ->  ( (Λ `  A )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^ k
 ) ) )
 
Theoremvmappw 20354 Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (Λ `  ( P ^ K ) )  =  ( log `  P ) )
 
Theoremvmaprm 20355 Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( P  e.  Prime  ->  (Λ `  P )  =  ( log `  P ) )
 
Theoremvmacl 20356 Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  NN  ->  (Λ `  A )  e.  RR )
 
Theoremvmaf 20357 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |- Λ : NN --> RR
 
Theoremefvmacl 20358 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  NN  ->  ( exp `  (Λ `  A ) )  e. 
 NN )
 
Theoremvmage0 20359 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  NN  ->  0  <_  (Λ `  A ) )
 
Theoremchpval 20360* Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  (
 1 ... ( |_ `  A ) ) (Λ `  n ) )
 
Theoremchpf 20361 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |- ψ : RR --> RR
 
Theoremchpcl 20362 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  (ψ `  A )  e.  RR )
 
Theoremefchpcl 20363 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  ( exp `  (ψ `  A ) )  e. 
 NN )
 
Theoremchpge0 20364 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
 |-  ( A  e.  RR  ->  0  <_  (ψ `  A ) )
 
Theoremppival 20365 Value of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  RR  ->  (π `  A )  =  ( # `  (
 ( 0 [,] A )  i^i  Prime ) ) )
 
Theoremppival2 20366 Value of the prime pi function. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  ZZ  ->  (π `  A )  =  ( # `  (
 ( 2 ... A )  i^i  Prime ) ) )
 
Theoremppival2g 20367 Value of the prime pi function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  2  e.  ( ZZ>=
 `  M ) ) 
 ->  (π `  A )  =  ( # `  (
 ( M ... A )  i^i  Prime ) ) )
 
Theoremppif 20368 Domain and range of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |- π : RR --> NN0
 
Theoremppicl 20369 Real closure of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  RR  ->  (π `  A )  e. 
 NN0 )
 
Theoremmuval 20370* The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  NN  ->  ( mmu `  A )  =  if ( E. p  e.  Prime  ( p ^ 2 ) 
 ||  A ,  0 ,  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  A } ) ) ) )
 
Theoremmuval1 20371 The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>=
 `  2 )  /\  ( P ^ 2 ) 
 ||  A )  ->  ( mmu `  A )  =  0 )
 
Theoremmuval2 20372* The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
 )  ->  ( mmu `  A )  =  (
 -u 1 ^ ( # `
  { p  e. 
 Prime  |  p  ||  A } ) ) )
 
Theoremisnsqf 20373* Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( A  e.  NN  ->  ( ( mmu `  A )  =  0  <->  E. p  e.  Prime  ( p ^ 2 )  ||  A ) )
 
Theoremissqf 20374* Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( A  e.  NN  ->  ( ( mmu `  A )  =/=  0  <->  A. p  e.  Prime  ( p  pCnt  A )  <_ 
 1 ) )
 
Theoremsqfpc 20375 The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015.)
 |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0  /\  P  e.  Prime )  ->  ( P  pCnt  A ) 
 <_  1 )
 
Theoremdvdssqf 20376 A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)
 |-  ( ( A  e.  NN  /\  B  e.  NN  /\  B  ||  A )  ->  ( ( mmu `  A )  =/=  0  ->  ( mmu `  B )  =/=  0 ) )
 
Theoremsqf11 20377* A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)
 |-  ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0 ) )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  ||  A  <->  p  ||  B ) ) )
 
Theoremmuf 20378 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |- 
 mmu : NN --> ZZ
 
Theoremmucl 20379 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  NN  ->  ( mmu `  A )  e.  ZZ )
 
Theoremsgmval 20380* The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  NN )  ->  ( A  sigma  B )  =  sum_ k  e.  { p  e.  NN  |  p  ||  B }  ( k  ^ c  A ) )
 
Theoremsgmval2 20381* The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  sigma  B )  =  sum_ k  e.  { p  e.  NN  |  p  ||  B }  ( k ^ A ) )
 
Theorem0sgm 20382* The value of the sum-of-divisors function, usually denoted σ<SUB>0</SUB>(<i>n</i>). (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( A  e.  NN  ->  ( 0  sigma  A )  =  ( # `  { p  e.  NN  |  p  ||  A } ) )
 
Theoremsgmf 20383 The divisor function is a function into the complex numbers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
 |- 
 sigma  : ( CC  X.  NN ) --> CC
 
Theoremsgmcl 20384 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  NN )  ->  ( A  sigma  B )  e.  CC )
 
Theoremsgmnncl 20385 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  sigma  B )  e.  NN )
 
Theoremmule1 20386 The Möbius function takes on values in magnitude at most  1. (Together with mucl 20379, this implies that it takes a value in  { -u 1 ,  0 ,  1 } for every natural number.) (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  NN  ->  ( abs `  ( mmu `  A ) ) 
 <_  1 )
 
Theoremchtfl 20387 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
 )
 
Theoremchpfl 20388 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( A  e.  RR  ->  (ψ `  ( |_ `  A ) )  =  (ψ `  A )
 )
 
Theoremppiprm 20389 The prime pi function at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime ) 
 ->  (π `  ( A  +  1 ) )  =  ( (π `  A )  +  1 ) )
 
Theoremppinprm 20390 The prime pi function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e. 
 Prime )  ->  (π `  ( A  +  1 )
 )  =  (π `  A ) )
 
Theoremchtprm 20391 The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime ) 
 ->  ( theta `  ( A  +  1 ) )  =  ( ( theta `  A )  +  ( log `  ( A  +  1 ) ) ) )
 
Theoremchtnprm 20392 The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e. 
 Prime )  ->  ( theta `  ( A  +  1 ) )  =  (
 theta `  A ) )
 
Theoremchpp1 20393 The second Chebyshev function at a successor. (Contributed by Mario Carneiro, 11-Apr-2016.)
 |-  ( A  e.  NN0  ->  (ψ `  ( A  +  1 ) )  =  ( (ψ `  A )  +  (Λ `  ( A  +  1 )
 ) ) )
 
Theoremchtwordi 20394 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  <_  ( theta `  B )
 )
 
Theoremchpwordi 20395 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (ψ `  A )  <_  (ψ `  B )
 )
 
Theoremchtdif 20396* The difference of the Chebyshev function at two points sums the logarithms of the primes in an interval. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ( theta `  N )  -  ( theta `  M ) )  =  sum_ p  e.  ( ( ( M  +  1 )
 ... N )  i^i 
 Prime ) ( log `  p ) )
 
Theoremefchtdvds 20397 The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A ) ) 
 ||  ( exp `  ( theta `  B ) ) )
 
Theoremppifl 20398 The prime pi function does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  RR  ->  (π `  ( |_ `  A ) )  =  (π `  A ) )
 
Theoremppip1le 20399 The prime pi function cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  ( A  e.  RR  ->  (π `  ( A  +  1 ) )  <_  ( (π `  A )  +  1 ) )
 
Theoremppiwordi 20400 The prime pi function is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  <_  (π
 `  B ) )
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