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Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchpdifbndlem2 21201* Lemma for chpdifbnd 21202. (Contributed by Mario Carneiro, 25-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  1 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  A. z  e.  ( 1 [,)  +oo ) ( abs `  ( ( ( ( (ψ `  z )  x.  ( log `  z
 ) )  +  sum_ m  e.  ( 1 ... ( |_ `  z
 ) ) ( (Λ `  m )  x.  (ψ `  ( z  /  m ) ) ) ) 
 /  z )  -  ( 2  x.  ( log `  z ) ) ) )  <_  B )   &    |-  C  =  ( ( B  x.  ( A  +  1 ) )  +  ( ( 2  x.  A )  x.  ( log `  A ) ) )   =>    |-  ( ph  ->  E. c  e.  RR+  A. x  e.  ( 1 (,)  +oo ) A. y  e.  ( x [,] ( A  x.  x ) ) ( (ψ `  y )  -  (ψ `  x )
 )  <_  ( (
 2  x.  ( y  -  x ) )  +  ( c  x.  ( x  /  ( log `  x ) ) ) ) )
 
Theoremchpdifbnd 21202* A bound on the difference of nearby ψ values. Theorem 10.5.2 of [Shapiro], p. 427. (Contributed by Mario Carneiro, 25-May-2016.)
 |-  ( ( A  e.  RR  /\  1  <_  A )  ->  E. c  e.  RR+  A. x  e.  ( 1 (,)  +oo ) A. y  e.  ( x [,] ( A  x.  x ) ) ( (ψ `  y
 )  -  (ψ `  x ) )  <_  ( ( 2  x.  ( y  -  x ) )  +  (
 c  x.  ( x 
 /  ( log `  x ) ) ) ) )
 
Theoremlogdivbnd 21203* A bound on a sum of logs, used in pntlemk 21253. This is not as precise as logdivsum 21180 in its asymptotic behavior, but it is valid for all  N and does not require a limit value. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( N  e.  NN  -> 
 sum_ n  e.  (
 1 ... N ) ( ( log `  n )  /  n )  <_  ( ( ( ( log `  N )  +  1 ) ^
 2 )  /  2
 ) )
 
Theoremselberg3lem1 21204* Introduce a log weighting on the summands of  sum_ m  x.  n  <_  x , Λ ( m )Λ ( n ), the core of selberg2 21198 (written here as  sum_ n  <_  x , Λ ( n )ψ (
x  /  n )). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. y  e.  ( 1 [,)  +oo ) ( abs `  ( ( sum_ k  e.  ( 1 ... ( |_ `  y ) ) ( (Λ `  k
 )  x.  ( log `  k ) )  -  ( (ψ `  y )  x.  ( log `  y
 ) ) )  /  y ) )  <_  A )   =>    |-  ( ph  ->  ( x  e.  ( 1 (,)  +oo )  |->  ( ( ( ( 2  /  ( log `  x )
 )  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n ) ) )  -  sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) 
 /  x ) )  e.  O ( 1 ) )
 
Theoremselberg3lem2 21205* Lemma for selberg3 21206. Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( ( ( 2 
 /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n ) ) )  -  sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) 
 /  x ) )  e.  O ( 1 )
 
Theoremselberg3 21206* Introduce a log weighting on the summands of  sum_ m  x.  n  <_  x , Λ ( m )Λ ( n ), the core of selberg2 21198 (written here as  sum_ n  <_  x , Λ ( n )ψ (
x  /  n )). Equation 10.6.7 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  ( ( 2 
 /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n ) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O ( 1 )
 
Theoremselberg4lem1 21207* Lemma for selberg4 21208. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. y  e.  ( 1 [,)  +oo ) ( abs `  ( ( sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( y  /  i ) ) ) )  /  y )  -  ( 2  x.  ( log `  y
 ) ) ) ) 
 <_  A )   =>    |-  ( ph  ->  ( x  e.  ( 1 (,)  +oo )  |->  ( (
 sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  sum_ m  e.  ( 1 ... ( |_ `  ( x  /  n ) ) ) ( (Λ `  m )  x.  ( ( log `  m )  +  (ψ `  ( ( x  /  n )  /  m ) ) ) ) )  /  ( x  x.  ( log `  x ) ) )  -  ( log `  x )
 ) )  e.  O ( 1 ) )
 
Theoremselberg4 21208* The Selberg symmetry formula for products of three primes, instead of two. The sum here can also be written in the symmetric form  sum_ i j k  <_  x , Λ ( i )Λ ( j )Λ ( k ); we eliminate one of the nested sums by using the definition of ψ ( x )  =  sum_ k  <_  x , Λ ( k ). This statement can thus equivalently be written ψ
( x ) log
^ 2 ( x )  =  2 sum_ i
j k  <_  x , Λ ( i )Λ (
j )Λ ( k )  +  O ( x log x ). Equation 10.4.23 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( ( (ψ `  x )  x.  ( log `  x ) )  -  ( ( 2 
 /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  sum_ m  e.  (
 1 ... ( |_ `  ( x  /  n ) ) ) ( (Λ `  m )  x.  (ψ `  (
 ( x  /  n )  /  m ) ) ) ) ) ) 
 /  x ) )  e.  O ( 1 )
 
Theorempntrval 21209* Define the residual of the second Chebyshev function. The goal is to have  R ( x )  e.  o ( x ), or  R ( x )  /  x  ~~> r  0. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  ( A  e.  RR+  ->  ( R `  A )  =  ( (ψ `  A )  -  A ) )
 
Theorempntrf 21210 Functionality of the residual. Lemma for pnt 21261. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  R : RR+ --> RR
 
Theorempntrmax 21211* There is a bound on the residual valid for all  x. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |- 
 E. c  e.  RR+  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  c
 
Theorempntrsumo1 21212* A bound on a sum over  R. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  ( x  e.  RR  |->  sum_
 n  e.  ( 1
 ... ( |_ `  x ) ) ( ( R `  n ) 
 /  ( n  x.  ( n  +  1
 ) ) ) )  e.  O ( 1 )
 
Theorempntrsumbnd 21213* A bound on a sum over  R. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |- 
 E. c  e.  RR+  A. m  e.  ZZ  ( abs `  sum_ n  e.  (
 1 ... m ) ( ( R `  n )  /  ( n  x.  ( n  +  1
 ) ) ) ) 
 <_  c
 
Theorempntrsumbnd2 21214* A bound on a sum over  R. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |- 
 E. c  e.  RR+  A. k  e.  NN  A. m  e.  ZZ  ( abs `  sum_ n  e.  (
 k ... m ) ( ( R `  n )  /  ( n  x.  ( n  +  1
 ) ) ) ) 
 <_  c
 
Theoremselbergr 21215* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  ( x  e.  RR+  |->  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
 )  x.  ( R `
  ( x  /  d ) ) ) )  /  x ) )  e.  O ( 1 )
 
Theoremselberg3r 21216* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.8 of [Shapiro], p. 429. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( ( ( R `
  x )  x.  ( log `  x ) )  +  (
 ( 2  /  ( log `  x ) )  x.  sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( ( (Λ `  n )  x.  ( R `  ( x  /  n ) ) )  x.  ( log `  n ) ) ) )  /  x ) )  e.  O ( 1 )
 
Theoremselberg4r 21217* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.11 of [Shapiro], p. 430. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( ( ( R `
  x )  x.  ( log `  x ) )  -  (
 ( 2  /  ( log `  x ) )  x.  sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  sum_ m  e.  ( 1 ... ( |_ `  ( x  /  n ) ) ) ( (Λ `  m )  x.  ( R `  ( ( x  /  n )  /  m ) ) ) ) ) )  /  x ) )  e.  O ( 1 )
 
Theoremselberg34r 21218* The sum of selberg3r 21216 and selberg4r 21217. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( ( ( R `
  x )  x.  ( log `  x ) )  -  ( sum_ n  e.  ( 1
 ... ( |_ `  x ) ) ( ( R `  ( x 
 /  n ) )  x.  ( sum_ m  e.  { y  e.  NN  |  y  ||  n }  ( (Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  -  ( (Λ `  n )  x.  ( log `  n ) ) ) ) 
 /  ( log `  x ) ) )  /  x ) )  e.  O ( 1 )
 
Theorempntsval 21219* Define the "Selberg function", whose asymptotic behavior is the content of selberg 21195. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   =>    |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  n )  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
 
Theorempntsf 21220* Functionality of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   =>    |-  S : RR --> RR
 
Theoremselbergs 21221* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   =>    |-  ( x  e.  RR+  |->  ( ( ( S `
  x )  /  x )  -  (
 2  x.  ( log `  x ) ) ) )  e.  O ( 1 )
 
Theoremselbergsb 21222* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   =>    |- 
 E. c  e.  RR+  A. x  e.  ( 1 [,)  +oo ) ( abs `  ( ( ( S `
  x )  /  x )  -  (
 2  x.  ( log `  x ) ) ) )  <_  c
 
Theorempntsval2 21223* The Selberg function can be expressed using the convolution product of the von Mangoldt function with itself. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   =>    |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( ( (Λ `  n )  x.  ( log `  n ) )  +  sum_ m  e.  { y  e.  NN  |  y  ||  n }  ( (Λ `  m )  x.  (Λ `  ( n  /  m ) ) ) ) )
 
Theorempntrlog2bndlem1 21224* The sum of selberg3r 21216 and selberg4r 21217. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   &    |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( ( ( abs `  ( R `  x ) )  x.  ( log `  x ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( R `  ( x  /  n ) ) )  x.  ( ( S `
  n )  -  ( S `  ( n  -  1 ) ) ) )  /  ( log `  x ) ) )  /  x ) )  e.  <_ O ( 1 )
 
Theorempntrlog2bndlem2 21225* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   &    |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. y  e.  RR+  (ψ `  y )  <_  ( A  x.  y ) )   =>    |-  ( ph  ->  ( x  e.  ( 1 (,)  +oo )  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( n  x.  ( abs `  ( ( R `
  ( x  /  ( n  +  1
 ) ) )  -  ( R `  ( x 
 /  n ) ) ) ) )  /  ( x  x.  ( log `  x ) ) ) )  e.  O ( 1 ) )
 
Theorempntrlog2bndlem3 21226* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   &    |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. y  e.  ( 1 [,)  +oo ) ( abs `  ( ( ( S `
  y )  /  y )  -  (
 2  x.  ( log `  y ) ) ) )  <_  A )   =>    |-  ( ph  ->  ( x  e.  ( 1 (,)  +oo )  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( abs `  ( R `  ( x  /  n ) ) )  -  ( abs `  ( R `  ( x  /  ( n  +  1 ) ) ) ) )  x.  (
 ( S `  n )  -  ( 2  x.  ( n  x.  ( log `  n ) ) ) ) )  /  ( x  x.  ( log `  x ) ) ) )  e.  O ( 1 ) )
 
Theorempntrlog2bndlem4 21227* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   &    |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  T  =  ( a  e.  RR  |->  if ( a  e.  RR+ ,  ( a  x.  ( log `  a ) ) ,  0 ) )   =>    |-  ( x  e.  (
 1 (,)  +oo )  |->  ( ( ( ( abs `  ( R `  x ) )  x.  ( log `  x ) )  -  ( ( 2 
 /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( R `  ( x  /  n ) ) )  x.  ( ( T `
  n )  -  ( T `  ( n  -  1 ) ) ) ) ) ) 
 /  x ) )  e.  <_ O ( 1 )
 
Theorempntrlog2bndlem5 21228* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   &    |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  T  =  ( a  e.  RR  |->  if ( a  e.  RR+ ,  ( a  x.  ( log `  a ) ) ,  0 ) )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  A. y  e.  RR+  ( abs `  ( ( R `  y )  /  y
 ) )  <_  B )   =>    |-  ( ph  ->  ( x  e.  ( 1 (,)  +oo )  |->  ( ( ( ( abs `  ( R `  x ) )  x.  ( log `  x ) )  -  (
 ( 2  /  ( log `  x ) )  x.  sum_ n  e.  (
 1 ... ( |_ `  x ) ) ( ( abs `  ( R `  ( x  /  n ) ) )  x.  ( log `  n ) ) ) ) 
 /  x ) )  e.  <_ O ( 1 ) )
 
Theorempntrlog2bndlem6a 21229* Lemma for pntrlog2bndlem6 21230. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   &    |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  T  =  ( a  e.  RR  |->  if ( a  e.  RR+ ,  ( a  x.  ( log `  a ) ) ,  0 ) )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  A. y  e.  RR+  ( abs `  ( ( R `  y )  /  y
 ) )  <_  B )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1 
 <_  A )   =>    |-  ( ( ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  ( 1
 ... ( |_ `  x ) )  =  (
 ( 1 ... ( |_ `  ( x  /  A ) ) )  u.  ( ( ( |_ `  ( x 
 /  A ) )  +  1 ) ... ( |_ `  x ) ) ) )
 
Theorempntrlog2bndlem6 21230* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
 )  x.  ( ( log `  i )  +  (ψ `  ( a  /  i ) ) ) ) )   &    |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  T  =  ( a  e.  RR  |->  if ( a  e.  RR+ ,  ( a  x.  ( log `  a ) ) ,  0 ) )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  A. y  e.  RR+  ( abs `  ( ( R `  y )  /  y
 ) )  <_  B )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1 
 <_  A )   =>    |-  ( ph  ->  ( x  e.  ( 1 (,)  +oo )  |->  ( ( ( ( abs `  ( R `  x ) )  x.  ( log `  x ) )  -  (
 ( 2  /  ( log `  x ) )  x.  sum_ n  e.  (
 1 ... ( |_ `  ( x  /  A ) ) ) ( ( abs `  ( R `  ( x  /  n ) ) )  x.  ( log `  n ) ) ) )  /  x ) )  e.  <_ O ( 1 ) )
 
Theorempntrlog2bnd 21231* A bound on  R ( x ) log ^ 2 ( x ). Equation 10.6.15 of [Shapiro], p. 431. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |-  ( ( A  e.  RR  /\  1  <_  A )  ->  E. c  e.  RR+  A. x  e.  ( 1 (,)  +oo ) ( ( ( ( abs `  ( R `  x ) )  x.  ( log `  x ) )  -  (
 ( 2  /  ( log `  x ) )  x.  sum_ n  e.  (
 1 ... ( |_ `  ( x  /  A ) ) ) ( ( abs `  ( R `  ( x  /  n ) ) )  x.  ( log `  n ) ) ) )  /  x ) 
 <_  c )
 
Theorempntpbnd1a 21232* Lemma for pntpbnd 21235. (Contributed by Mario Carneiro, 11-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )   &    |-  X  =  ( exp `  (
 2  /  E )
 )   &    |-  ( ph  ->  Y  e.  ( X (,)  +oo ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( Y  <  N 
 /\  N  <_  ( K  x.  Y ) ) )   &    |-  ( ph  ->  ( abs `  ( R `  N ) )  <_  ( abs `  ( ( R `  ( N  +  1 ) )  -  ( R `  N ) ) ) )   =>    |-  ( ph  ->  ( abs `  ( ( R `  N )  /  N ) )  <_  E )
 
Theorempntpbnd1 21233* Lemma for pntpbnd 21235. (Contributed by Mario Carneiro, 11-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )   &    |-  X  =  ( exp `  (
 2  /  E )
 )   &    |-  ( ph  ->  Y  e.  ( X (,)  +oo ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. i  e.  NN  A. j  e.  ZZ  ( abs `  sum_ y  e.  (
 i ... j ) ( ( R `  y
 )  /  ( y  x.  ( y  +  1 ) ) ) ) 
 <_  A )   &    |-  C  =  ( A  +  2 )   &    |-  ( ph  ->  K  e.  ( ( exp `  ( C  /  E ) ) [,)  +oo ) )   &    |-  ( ph  ->  -.  E. y  e.  NN  ( ( Y  <  y  /\  y  <_  ( K  x.  Y ) )  /\  ( abs `  ( ( R `  y )  /  y
 ) )  <_  E ) )   =>    |-  ( ph  ->  sum_ n  e.  ( ( ( |_ `  Y )  +  1 ) ... ( |_ `  ( K  x.  Y ) ) ) ( abs `  ( ( R `  n )  /  ( n  x.  ( n  +  1 )
 ) ) )  <_  A )
 
Theorempntpbnd2 21234* Lemma for pntpbnd 21235. (Contributed by Mario Carneiro, 11-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )   &    |-  X  =  ( exp `  (
 2  /  E )
 )   &    |-  ( ph  ->  Y  e.  ( X (,)  +oo ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. i  e.  NN  A. j  e.  ZZ  ( abs `  sum_ y  e.  (
 i ... j ) ( ( R `  y
 )  /  ( y  x.  ( y  +  1 ) ) ) ) 
 <_  A )   &    |-  C  =  ( A  +  2 )   &    |-  ( ph  ->  K  e.  ( ( exp `  ( C  /  E ) ) [,)  +oo ) )   &    |-  ( ph  ->  -.  E. y  e.  NN  ( ( Y  <  y  /\  y  <_  ( K  x.  Y ) )  /\  ( abs `  ( ( R `  y )  /  y
 ) )  <_  E ) )   =>    |- 
 -.  ph
 
Theorempntpbnd 21235* Lemma for pnt 21261. Establish smallness of  R at a point. Lemma 10.6.1 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |- 
 E. c  e.  RR+  A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  ( ( exp `  (
 c  /  e )
 ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. n  e.  NN  ( ( y  <  n  /\  n  <_  ( k  x.  y
 ) )  /\  ( abs `  ( ( R `
  n )  /  n ) )  <_  e )
 
Theorempntibndlem1 21236 Lemma for pntibnd 21240. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  L  =  ( ( 1  /  4
 )  /  ( A  +  3 ) )   =>    |-  ( ph  ->  L  e.  ( 0 (,) 1
 ) )
 
Theorempntibndlem2a 21237* Lemma for pntibndlem2 21238. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  L  =  ( ( 1  /  4
 )  /  ( A  +  3 ) )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  (
 ( R `  x )  /  x ) ) 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  K  =  ( exp `  ( B  /  ( E  /  2
 ) ) )   &    |-  C  =  ( ( 2  x.  B )  +  ( log `  2 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1
 ) )   &    |-  ( ph  ->  Z  e.  RR+ )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ( ph  /\  u  e.  ( N [,] ( ( 1  +  ( L  x.  E ) )  x.  N ) ) ) 
 ->  ( u  e.  RR  /\  N  <_  u  /\  u  <_  ( ( 1  +  ( L  x.  E ) )  x.  N ) ) )
 
Theorempntibndlem2 21238* Lemma for pntibnd 21240. The main work, after eliminating all the quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  L  =  ( ( 1  /  4
 )  /  ( A  +  3 ) )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  (
 ( R `  x )  /  x ) ) 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  K  =  ( exp `  ( B  /  ( E  /  2
 ) ) )   &    |-  C  =  ( ( 2  x.  B )  +  ( log `  2 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1
 ) )   &    |-  ( ph  ->  Z  e.  RR+ )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  ( 1 (,)  +oo ) A. y  e.  ( x [,] ( 2  x.  x ) ) ( (ψ `  y )  -  (ψ `  x )
 )  <_  ( (
 2  x.  ( y  -  x ) )  +  ( T  x.  ( x  /  ( log `  x ) ) ) ) )   &    |-  X  =  ( ( exp `  ( T  /  ( E  / 
 4 ) ) )  +  Z )   &    |-  ( ph  ->  M  e.  (
 ( exp `  ( C  /  E ) ) [,)  +oo ) )   &    |-  ( ph  ->  Y  e.  ( X (,)  +oo ) )   &    |-  ( ph  ->  ( ( Y  <  N  /\  N  <_  ( ( M  /  2 )  x.  Y ) )  /\  ( abs `  ( ( R `  N )  /  N ) )  <_  ( E  /  2
 ) ) )   =>    |-  ( ph  ->  E. z  e.  RR+  ( ( Y  <  z  /\  ( ( 1  +  ( L  x.  E ) )  x.  z
 )  <  ( M  x.  Y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
 
Theorempntibndlem3 21239* Lemma for pntibnd 21240. Package up pntibndlem2 21238 in quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  L  =  ( ( 1  /  4
 )  /  ( A  +  3 ) )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  (
 ( R `  x )  /  x ) ) 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  K  =  ( exp `  ( B  /  ( E  /  2
 ) ) )   &    |-  C  =  ( ( 2  x.  B )  +  ( log `  2 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1
 ) )   &    |-  ( ph  ->  Z  e.  RR+ )   &    |-  ( ph  ->  A. m  e.  ( K [,)  +oo ) A. v  e.  ( Z (,)  +oo ) E. i  e.  NN  ( ( v  < 
 i  /\  i  <_  ( m  x.  v ) )  /\  ( abs `  ( ( R `  i )  /  i
 ) )  <_  ( E  /  2 ) ) )   =>    |-  ( ph  ->  E. x  e.  RR+  A. k  e.  (
 ( exp `  ( C  /  E ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
 
Theorempntibnd 21240* Lemma for pnt 21261. Establish smallness of  R on an interval. Lemma 10.6.2 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |- 
 E. c  e.  RR+  E. l  e.  ( 0 (,) 1 ) A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  (
 ( exp `  ( c  /  e ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( l  x.  e ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( l  x.  e
 ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  e )
 
Theorempntlemd 21241 Lemma for pnt 21261. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  A is C^*,  B is c1,  L is λ,  D is c2, and  F is c3. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   =>    |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
 
Theorempntlemc 21242* Lemma for pnt 21261. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  U is α,  E is ε, and  K is K. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   =>    |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1
 )  /\  1  <  K 
 /\  ( U  -  E )  e.  RR+ )
 ) )
 
Theorempntlema 21243* Lemma for pnt 21261. Closure for the constants used in the proof. The mammoth expression  W is a number large enough to satisfy all the lower bounds needed for  Z. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  Y is x2,  X is x1,  C is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and  W is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   =>    |-  ( ph  ->  W  e.  RR+ )
 
Theorempntlemb 21244* Lemma for pnt 21261. Unpack all the lower bounds contained in  W, in the form they will be used. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  Z is x. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   =>    |-  ( ph  ->  ( Z  e.  RR+  /\  (
 1  <  Z  /\  _e  <_  ( sqr `  Z )  /\  ( sqr `  Z )  <_  ( Z  /  Y ) )  /\  ( ( 4  /  ( L  x.  E ) )  <_  ( sqr `  Z )  /\  (
 ( ( log `  X )  /  ( log `  K ) )  +  2
 )  <_  ( (
 ( log `  Z )  /  ( log `  K ) )  /  4
 )  /\  ( ( U  x.  3 )  +  C )  <_  ( ( ( U  -  E )  x.  ( ( L  x.  ( E ^
 2 ) )  /  (; 3 2  x.  B ) ) )  x.  ( log `  Z ) ) ) ) )
 
Theorempntlemg 21245* Lemma for pnt 21261. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  M is j^* and  N is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   =>    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  ( ( ( log `  Z )  /  ( log `  K ) ) 
 /  4 )  <_  ( N  -  M ) ) )
 
Theorempntlemh 21246* Lemma for pnt 21261. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   =>    |-  ( ( ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J )  /\  ( K ^ J )  <_  ( sqr `  Z )
 ) )
 
Theorempntlemn 21247* Lemma for pnt 21261. The "naive" base bound, which we will slightly improve. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   =>    |-  ( ( ph  /\  ( J  e.  NN  /\  J  <_  ( Z  /  Y ) ) )  -> 
 0  <_  ( (
 ( U  /  J )  -  ( abs `  (
 ( R `  ( Z  /  J ) ) 
 /  Z ) ) )  x.  ( log `  J ) ) )
 
Theorempntlemq 21248* Lemma for pntlemj 21250. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  I  C_  O )
 
Theorempntlemr 21249* Lemma for pntlemj 21250. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  E ) 
 /  8 )  x.  ( log `  Z ) ) )  <_  ( ( # `  I
 )  x.  ( ( U  -  E )  x.  ( ( log `  ( Z  /  V ) )  /  ( Z  /  V ) ) ) ) )
 
Theorempntlemj 21250* Lemma for pnt 21261. The induction step. Using pntibnd 21240, we find an interval in  K ^ J ... K ^ ( J  + 
1 ) which is sufficiently large and has a much smaller value,  R ( z )  / 
z  <_  E (instead of our original bound 
R ( z )  /  z  <_  U). (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  E ) 
 /  8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  ( ( R `
  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemi 21251* Lemma for pnt 21261. Eliminate some assumptions from pntlemj 21250. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   =>    |-  ( ( ph  /\  J  e.  ( M..^ N ) )  ->  ( ( U  -  E )  x.  ( ( ( L  x.  E )  / 
 8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  (
 ( R `  ( Z  /  n ) ) 
 /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemf 21252* Lemma for pnt 21261. Add up the pieces in pntlemi 21251 to get an estimate slightly better than the naive lower bound  0. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  ( E ^ 2 ) ) 
 /  (; 3 2  x.  B ) )  x.  (
 ( log `  Z ) ^ 2 ) ) )  <_  sum_ n  e.  ( 1 ... ( |_ `  ( Z  /  Y ) ) ) ( ( ( U 
 /  n )  -  ( abs `  ( ( R `  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemk 21253* Lemma for pnt 21261. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   =>    |-  ( ph  ->  (
 2  x.  sum_ n  e.  ( 1 ... ( |_ `  ( Z  /  Y ) ) ) ( ( U  /  n )  x.  ( log `  n ) ) )  <_  ( ( U  x.  ( ( log `  Z )  +  3 ) )  x.  ( log `  Z ) ) )
 
Theorempntlemo 21254* Lemma for pnt 21261. Combine all the estimates to establish a smaller eventual bound on  R ( Z )  /  Z. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  A. z  e.  ( 1 (,)  +oo ) ( ( ( ( abs `  ( R `  z ) )  x.  ( log `  z
 ) )  -  (
 ( 2  /  ( log `  z ) )  x.  sum_ i  e.  (
 1 ... ( |_ `  (
 z  /  Y )
 ) ) ( ( abs `  ( R `  ( z  /  i
 ) ) )  x.  ( log `  i
 ) ) ) ) 
 /  z )  <_  C )   =>    |-  ( ph  ->  ( abs `  ( ( R `
  Z )  /  Z ) )  <_  ( U  -  ( F  x.  ( U ^
 3 ) ) ) )
 
Theorempntleme 21255* Lemma for pnt 21261. Package up pntlemo 21254 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  (
 ( R `  z
 )  /  z )
 )  <_  U )   &    |-  ( ph  ->  A. k  e.  ( K [,)  +oo ) A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  A. z  e.  ( 1 (,)  +oo ) ( ( ( ( abs `  ( R `  z ) )  x.  ( log `  z
 ) )  -  (
 ( 2  /  ( log `  z ) )  x.  sum_ i  e.  (
 1 ... ( |_ `  (
 z  /  Y )
 ) ) ( ( abs `  ( R `  ( z  /  i
 ) ) )  x.  ( log `  i
 ) ) ) ) 
 /  z )  <_  C )   =>    |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
 ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
 
Theorempntlem3 21256* Lemma for pnt 21261. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  T  =  { t  e.  ( 0 [,] A )  |  E. y  e.  RR+  A. z  e.  (
 y [,)  +oo ) ( abs `  ( ( R `  z )  /  z ) )  <_  t }   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  u  e.  T ) 
 ->  ( u  -  ( C  x.  ( u ^
 3 ) ) )  e.  T )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1 )
 
Theorempntlemp 21257* Lemma for pnt 21261. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  ( ( exp `  ( B  /  e ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  e ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  e
 ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  e ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   =>    |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
 ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
 
Theorempntleml 21258* Lemma for pnt 21261. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  ( ( exp `  ( B  /  e ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  e ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  e
 ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  e ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1 )
 
Theorempnt3 21259 The Prime Number Theorem, version 3: the second Chebyshev function tends asymptotically to  x. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1
 
Theorempnt2 21260 The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to  x. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  RR+  |->  ( ( theta `  x )  /  x ) )  ~~> r  1
 
Theorempnt 21261 The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( (π `  x )  /  ( x  /  ( log `  x ) ) ) )  ~~> r  1
 
13.4.13  Ostrowski's theorem
 
Theoremabvcxp 21262* Raising an absolute value to a power less than one yields another absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( x  e.  B  |->  ( ( F `  x )  ^ c  S ) )   =>    |-  ( ( F  e.  A  /\  S  e.  (
 0 (,] 1 ) ) 
 ->  G  e.  A )
 
Theorempadicfval 21263* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( P  e.  Prime  ->  ( J `  P )  =  ( x  e. 
 QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) ) ) )
 
Theorempadicval 21264* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  X  e.  QQ )  ->  ( ( J `
  P ) `  X )  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  X ) ) ) )
 
Theoremostth2lem1 21265* Lemma for ostth2 21284, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 21284. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, 
n  e.  o ( A ^ n ) for any 
1  <  A. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( A ^ n )  <_  ( n  x.  B ) )   =>    |-  ( ph  ->  A  <_  1 )
 
Theoremqrngbas 21266 The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |- 
 QQ  =  ( Base `  Q )
 
Theoremqdrng 21267 The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  Q  e.  DivRing
 
Theoremqrng0 21268 The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  0  =  ( 0g
 `  Q )
 
Theoremqrng1 21269 The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  1  =  ( 1r
 `  Q )
 
Theoremqrngneg 21270 The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  ( X  e.  QQ  ->  ( ( inv g `  Q ) `  X )  =  -u X )
 
Theoremqrngdiv 21271 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  ( ( X  e.  QQ  /\  Y  e.  QQ  /\  Y  =/=  0 ) 
 ->  ( X (/r `  Q ) Y )  =  ( X  /  Y ) )
 
Theoremqabvle 21272 By using induction on  N, we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( ( F  e.  A  /\  N  e.  NN0 )  ->  ( F `  N )  <_  N )
 
Theoremqabvexp 21273 Induct the product rule abvmul 15872 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( ( F  e.  A  /\  M  e.  QQ  /\  N  e.  NN0 )  ->  ( F `  ( M ^ N ) )  =  ( ( F `
  M ) ^ N ) )
 
Theoremostthlem1 21274* Lemma for ostth 21286. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ( ph  /\  n  e.  ( ZZ>= `  2 )
 )  ->  ( F `  n )  =  ( G `  n ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremostthlem2 21275* Lemma for ostth 21286. Refine ostthlem1 21274 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ( ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremqabsabv 21276 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( abs  |`  QQ )  e.  A
 
Theorempadicabv 21277* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  F  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( N ^ ( P  pCnt  x ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  N  e.  (
 0 (,) 1 ) ) 
 ->  F  e.  A )
 
Theorempadicabvf 21278* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  J : Prime --> A
 
Theorempadicabvcxp 21279* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( y  e. 
 QQ  |->  ( ( ( J `  P ) `
  y )  ^ c  R ) )  e.  A )
 
Theoremostth1 21280* - Lemma for ostth 21286: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If  F is equal to  1 on the primes, then by complete induction and the multiplicative property abvmul 15872 of the absolute value,  F is equal to  1 on all the integers, and ostthlem1 21274 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
  n ) )   &    |-  ( ph  ->  A. n  e. 
 Prime  -.  ( F `  n )  <  1 )   =>    |-  ( ph  ->  F  =  K )
 
Theoremostth2lem2 21281* Lemma for ostth2 21284. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   =>    |-  ( ( ph  /\  X  e.  NN0  /\  Y  e.  ( 0 ... (
 ( M ^ X )  -  1 ) ) )  ->  ( F `  Y )  <_  (
 ( M  x.  X )  x.  ( T ^ X ) ) )
 
Theoremostth2lem3 21282* Lemma for ostth2 21284. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   &    |-  U  =  ( ( log `  N )  /  ( log `  M )
 )   =>    |-  ( ( ph  /\  X  e.  NN )  ->  (
 ( ( F `  N )  /  ( T  ^ c  U ) ) ^ X ) 
 <_  ( X  x.  (
 ( M  x.  T )  x.  ( U  +  1 ) ) ) )
 
Theoremostth2lem4 21283* Lemma for ostth2 21284. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   &    |-  U  =  ( ( log `  N )  /  ( log `  M )
 )   =>    |-  ( ph  ->  (
 1  <  ( F `  M )  /\  R  <_  S ) )
 
Theoremostth2 21284* - Lemma for ostth 21286: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   =>    |-  ( ph  ->  E. a  e.  ( 0 (,] 1
 ) F  =  ( y  e.  QQ  |->  ( ( abs `  y
 )  ^ c  a ) ) )
 
Theoremostth3 21285* - Lemma for ostth 21286: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
  n ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( F `  P )  <  1 )   &    |-  R  =  -u ( ( log `  ( F `  P ) )  /  ( log `  P ) )   &    |-  S  =  if (
 ( F `  P )  <_  ( F `  p ) ,  ( F `  p ) ,  ( F `  P ) )   =>    |-  ( ph  ->  E. a  e.  RR+  F  =  ( y  e.  QQ  |->  ( ( ( J `  P ) `  y
 )  ^ c  a ) ) )
 
Theoremostth 21286* Ostrowski's theorem, which classifies all absolute values on  QQ. Any such absolute value must either be the trivial absolute value  K, a constant exponent  0  <  a  <_  1 times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   =>    |-  ( F  e.  A  <->  ( F  =  K  \/  E. a  e.  ( 0 (,] 1 ) F  =  ( y  e. 
 QQ  |->  ( ( abs `  y )  ^ c  a ) )  \/ 
 E. a  e.  RR+  E. g  e.  ran  J  F  =  ( y  e.  QQ  |->  ( ( g `
  y )  ^ c  a ) ) ) )
 
PART 14  GRAPH THEORY



To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to Definitions in [Bollobas] p. 1-8. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic kinds of graphs:

TermReferenceDefinitionRemarks
(Undirected) Hypergraph df-uhgra 21288 an ordered pair  <. V ,  E >. of a set  V and a function  E into the powerset of  V ( ran  E  C_  ( ~P V )).
An element of  V is called "vertex", an element of  ran  E is called "edge", the function  E is called the "edge-function" .
In this most general definition of a graph, an "edge" may connect three or more vertices with each other, compare with the definition in Section I.1 in [Bollobas] p. 7.
Undirected multigraph df-umgra 21301 a graph  <. V ,  E >. such that  E is a function into the set of (proper or not proper) unordered pairs of  V.A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 3940).
According to the definition in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed".
Undirected simple graph with loops df-uslgra 21319 a graph  <. V ,  E >. such that  E is a one-to-one function into the set of (proper or not proper) unordered pairs of  V.This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph without loops (in short "simple graph") df-usgra 21320 a graph  <. V ,  E >. such that  E is a one-to-one function into the set of (proper) unordered pairs of  V.An ordered pair  <. V ,  E >. of two distinct sets  V and  E (the "usual" definition of a "graph", see, for example, the definition in Section I.1 in [Bollobas] p. 1) can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see ausisusgra 21333.
Finite graph---a graph  <. V ,  E >. with finite sets  V and  E.In simple graphs,  E is finite if  V is finite, see usgrafis 21382. The number of edges is limited by  ( n  _C  2 ) (or " n choose 2") with  n  =  ( # `  V ), see usgramaxsize 21449. Analogously, the number of edges of an undirected simple graph with loops is limited by  ( ( n  +  1 )  _C  2 ). In multigraphs, however,  E can be infinite although  V is finite.
Graph of finite size---a graph  <. V ,  E >. with finite set  E, i.e. with a finite number of edges.A graph can be of finite size although  V is infinite.


Terms and properties of graphs:
TermReferenceDefinitionRemarks
Edge joining (two) vertices --- An edge  e  e.  ran  E "joins" the vertices v1, v2, ... vn ( n  e.  NN) if  e = { v1, v2, ... vn }. If  n  =  1,  e = { v1 } is a "loop", if  n  =  2,  e = { v1 , v2 } is an egde as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge see definition in Section I.1 in [Bollobas] p. 1. If an edge  e  e.  ran  E joins the vertices v1, v2, ... vn ( n  e.  NN), then the vertices v1, v2, ... vn are called the "endvertices" of the edge  e.
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v1, v2, ... vn ( n  e.  NN) are "adjacent" if there is an edge e = { v1, v2, ... vn } joining these vertices. In this case, the vertices are "incident" with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or "connected" by the edge e.
(Two) Adjacent edges The edges e0, e1, ... en ( n  e.  NN) are "adjacent" if they have exactly one common endvertex. Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph see definition in Section I.1 in [Bollobas] p. 3 the "order" of a graph  <. V ,  E >. is the number of vertices in the graph ( ( # `  V )).
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 the "size" of a graph  <. V ,  E >. is the number of edges in the graph ( ( # `  E )).
Neighborhood of a vertex df-nbgra 21386 resp. definition in Section I.1 in [Bollobas] p. 3 A vertex connected with a vertex  v by an edge is called a "neighbor" of the vertex  v. The set of neighbors of a vertex  v is called the "neighborhood" (or "open neighborhood") of the vertex  v. The "closed neighborhood" is the union of the (open) neighborhood of the vertex  v with  { v }.
Degree of a vertex df-vdgr 21618 The "degree" of a vertex is the number of the edges having this vertex as endvertex. In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex usgravd0nedg 21636 A vertex is called "isolated" if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 21388 A vertex is called "universal" if it is connected with every other vertex of the graph by an edge, thus having degree  ( # `  V ).


Special kinds of graphs:
TermReferenceDefinitionRemarks
Complete graph df-cusgra 21387 A graph is called "complete" if each pair of vertices is connected by an edge. The size of a complete undirected simple graph of order  n is  ( n  _C  2 ) (or " n choose 2"), see cusgrasize 21440.
Empty graph umgra0 21313 and usgra0 21343 A graph is called "empty" if it has no edges.
Null graph usgra0v 21344 A graph is called the "null graph" if it has no vertices (and therefore also no edges).
Trivial graph usgra1v 21362 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngra 21610 resp. definition in Section I.1 in [Bollobas] p. 6 A graph is called "connected" if for each pair of vertices there is a path between these vertices.


For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.
 
14.1  Undirected graphs - basics
 
14.1.1  Undirected hypergraphs
 
Syntaxcuhg 21287 Extend class notation with undirected hypergraphs.
 class UHGrph
 
Definitiondf-uhgra 21288* Define the class of all undirected hypergraphs. An undirected hypergraph is a pair of a set and a function into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |- UHGrph  =  { <. v ,  e >.  |  e : dom  e
 --> ( ~P v  \  { (/) } ) }
 
Theoremreluhgra 21289 The class of all undirected hypergraphs is a relation. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |- 
 Rel UHGrph
 
Theoremuhgrav 21290 The classes of vertices and edges of an undirected hypergraph are sets. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  ( V  e.  _V  /\  E  e.  _V )
 )
 
Theoremisuhgra 21291 The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } )
 ) )
 
Theoremuhgraf 21292 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } )
 )
 
Theoremuhgrafun 21293 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  Fun 
 E )
 
Theoremuhgrass 21294 An edge is a subset of vertices, analogous to umgrass 21307. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( V UHGrph  E  /\  F  e.  dom  E )  ->  ( E `  F )  C_  V )
 
Theoremuhgraeq12d 21295 Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( V  =  W  /\  E  =  F )
 )  ->  ( V UHGrph  E  <->  W UHGrph  F ) )
 
Theoremuhgrares 21296 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 21312. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V UHGrph  E  ->  V UHGrph 
 ( E  |`  A ) )
 
Theoremuhgra0 21297 The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 21313. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V  e.  W  ->  V UHGrph  (/) )
 
Theoremuhgra0v 21298 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( (/) UHGrph  E  <->  E  =  (/) )
 
Theoremuhgraun 21299 If  <. V ,  E >. and  <. V ,  F >. are hypergraphs, then  <. V ,  E  u.  F >. is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting in two edges between two vertices), analogous to umgraun 21316. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UHGrph  E )   &    |-  ( ph  ->  V UHGrph  F )   =>    |-  ( ph  ->  V UHGrph 
 ( E  u.  F ) )
 
14.1.2  Undirected multigraphs
 
Syntaxcumg 21300 Extend class notation with undirected multigraphs.
 class UMGrph
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