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Theorem chtdif 23257
Description: 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.)
Assertion
Ref Expression
chtdif  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( theta `  N )  -  ( theta `  M )
)  =  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) ( log `  p
) )
Distinct variable groups:    M, p    N, p

Proof of Theorem chtdif
StepHypRef Expression
1 eluzelre 11093 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  RR )
2 chtval 23209 . . . . 5  |-  ( N  e.  RR  ->  ( theta `  N )  = 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p ) )
31, 2syl 16 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( theta `  N )  =  sum_ p  e.  ( ( 0 [,] N )  i^i 
Prime ) ( log `  p
) )
4 eluzel2 11088 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5 2z 10897 . . . . . . . . . 10  |-  2  e.  ZZ
6 ifcl 3981 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  2  e.  ZZ )  ->  if ( M  <_ 
2 ,  M , 
2 )  e.  ZZ )
74, 5, 6sylancl 662 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  if ( M  <_  2 ,  M ,  2 )  e.  ZZ )
85a1i 11 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  2  e.  ZZ )
94zred 10967 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  RR )
10 2re 10606 . . . . . . . . . 10  |-  2  e.  RR
11 min2 11391 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  2  e.  RR )  ->  if ( M  <_ 
2 ,  M , 
2 )  <_  2
)
129, 10, 11sylancl 662 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  if ( M  <_  2 ,  M ,  2 )  <_ 
2 )
13 eluz2 11089 . . . . . . . . 9  |-  ( 2  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) )  <->  ( if ( M  <_  2 ,  M ,  2 )  e.  ZZ  /\  2  e.  ZZ  /\  if ( M  <_  2 ,  M ,  2 )  <_  2 ) )
147, 8, 12, 13syl3anbrc 1180 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  2  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) ) )
15 ppisval2 23203 . . . . . . . 8  |-  ( ( N  e.  RR  /\  2  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) ) )  ->  (
( 0 [,] N
)  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  N ) )  i^i 
Prime ) )
161, 14, 15syl2anc 661 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
0 [,] N )  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  N
) )  i^i  Prime ) )
17 eluzelz 11092 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
18 flid 11914 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( |_ `  N )  =  N )
1917, 18syl 16 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( |_ `  N )  =  N )
2019oveq2d 6301 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  N ) )  =  ( if ( M  <_  2 ,  M ,  2 ) ... N ) )
2120ineq1d 3699 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  N ) )  i^i 
Prime )  =  (
( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i  Prime ) )
2216, 21eqtrd 2508 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
0 [,] N )  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i  Prime ) )
2322sumeq1d 13489 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  sum_ p  e.  ( ( 0 [,] N )  i^i  Prime ) ( log `  p
)  =  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) ( log `  p
) )
249ltp1d 10477 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <  ( M  +  1 ) )
25 fzdisj 11713 . . . . . . . . 9  |-  ( M  <  ( M  + 
1 )  ->  (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  (
( M  +  1 ) ... N ) )  =  (/) )
2624, 25syl 16 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  ( ( M  + 
1 ) ... N
) )  =  (/) )
2726ineq1d 3699 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  (
( M  +  1 ) ... N ) )  i^i  Prime )  =  ( (/)  i^i  Prime ) )
28 inindir 3716 . . . . . . 7  |-  ( ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  (
( M  +  1 ) ... N ) )  i^i  Prime )  =  ( ( ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)  i^i  Prime )  i^i  ( ( ( M  +  1 ) ... N )  i^i  Prime ) )
29 incom 3691 . . . . . . . 8  |-  ( (/)  i^i 
Prime )  =  ( Prime  i^i  (/) )
30 in0 3811 . . . . . . . 8  |-  ( Prime  i^i  (/) )  =  (/)
3129, 30eqtri 2496 . . . . . . 7  |-  ( (/)  i^i 
Prime )  =  (/)
3227, 28, 313eqtr3g 2531 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  Prime )  i^i  ( ( ( M  +  1 ) ... N )  i^i 
Prime ) )  =  (/) )
33 min1 11390 . . . . . . . . . . . 12  |-  ( ( M  e.  RR  /\  2  e.  RR )  ->  if ( M  <_ 
2 ,  M , 
2 )  <_  M
)
349, 10, 33sylancl 662 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  M
)  ->  if ( M  <_  2 ,  M ,  2 )  <_  M )
35 eluz2 11089 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) )  <->  ( if ( M  <_  2 ,  M ,  2 )  e.  ZZ  /\  M  e.  ZZ  /\  if ( M  <_  2 ,  M ,  2 )  <_  M ) )
367, 4, 34, 35syl3anbrc 1180 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) ) )
37 id 22 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( ZZ>= `  M )
)
38 elfzuzb 11683 . . . . . . . . . 10  |-  ( M  e.  ( if ( M  <_  2 ,  M ,  2 ) ... N )  <->  ( M  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) )  /\  N  e.  (
ZZ>= `  M ) ) )
3936, 37, 38sylanbrc 664 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( if ( M  <_ 
2 ,  M , 
2 ) ... N
) )
40 fzsplit 11712 . . . . . . . . 9  |-  ( M  e.  ( if ( M  <_  2 ,  M ,  2 ) ... N )  -> 
( if ( M  <_  2 ,  M ,  2 ) ... N )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  u.  (
( M  +  1 ) ... N ) ) )
4139, 40syl 16 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( if ( M  <_  2 ,  M ,  2 ) ... N )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  u.  ( ( M  + 
1 ) ... N
) ) )
4241ineq1d 3699 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime )  =  (
( ( if ( M  <_  2 ,  M ,  2 ) ... M )  u.  ( ( M  + 
1 ) ... N
) )  i^i  Prime ) )
43 indir 3746 . . . . . . 7  |-  ( ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  u.  (
( M  +  1 ) ... N ) )  i^i  Prime )  =  ( ( ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)  i^i  Prime )  u.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) )
4442, 43syl6eq 2524 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime )  =  (
( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime )  u.  (
( ( M  + 
1 ) ... N
)  i^i  Prime ) ) )
45 fzfid 12052 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( if ( M  <_  2 ,  M ,  2 ) ... N )  e. 
Fin )
46 inss1 3718 . . . . . . 7  |-  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... N
)  i^i  Prime )  C_  ( if ( M  <_ 
2 ,  M , 
2 ) ... N
)
47 ssfi 7741 . . . . . . 7  |-  ( ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  e.  Fin  /\  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime )  C_  ( if ( M  <_  2 ,  M ,  2 ) ... N ) )  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime )  e.  Fin )
4845, 46, 47sylancl 662 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime )  e.  Fin )
49 inss2 3719 . . . . . . . . . . 11  |-  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... N
)  i^i  Prime )  C_  Prime
50 simpr 461 . . . . . . . . . . 11  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )
5149, 50sseldi 3502 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  p  e.  Prime )
52 prmnn 14082 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
5351, 52syl 16 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  p  e.  NN )
5453nnrpd 11256 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  p  e.  RR+ )
5554relogcld 22833 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
5655recnd 9623 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
5732, 44, 48, 56fsumsplit 13528 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) ( log `  p
)  =  ( sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
)  +  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) ( log `  p
) ) )
5823, 57eqtrd 2508 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  sum_ p  e.  ( ( 0 [,] N )  i^i  Prime ) ( log `  p
)  =  ( sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
)  +  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) ( log `  p
) ) )
593, 58eqtrd 2508 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( theta `  N )  =  (
sum_ p  e.  (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  Prime ) ( log `  p
)  +  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) ( log `  p
) ) )
60 chtval 23209 . . . . 5  |-  ( M  e.  RR  ->  ( theta `  M )  = 
sum_ p  e.  (
( 0 [,] M
)  i^i  Prime ) ( log `  p ) )
619, 60syl 16 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( theta `  M )  =  sum_ p  e.  ( ( 0 [,] M )  i^i 
Prime ) ( log `  p
) )
62 ppisval2 23203 . . . . . . 7  |-  ( ( M  e.  RR  /\  2  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) ) )  ->  (
( 0 [,] M
)  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  M ) )  i^i 
Prime ) )
639, 14, 62syl2anc 661 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
0 [,] M )  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  M
) )  i^i  Prime ) )
64 flid 11914 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( |_ `  M )  =  M )
654, 64syl 16 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( |_ `  M )  =  M )
6665oveq2d 6301 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  M ) )  =  ( if ( M  <_  2 ,  M ,  2 ) ... M ) )
6766ineq1d 3699 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  M ) )  i^i 
Prime )  =  (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  Prime ) )
6863, 67eqtrd 2508 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
0 [,] M )  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  Prime ) )
6968sumeq1d 13489 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  sum_ p  e.  ( ( 0 [,] M )  i^i  Prime ) ( log `  p
)  =  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
) )
7061, 69eqtrd 2508 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( theta `  M )  =  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
) )
7159, 70oveq12d 6303 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( theta `  N )  -  ( theta `  M )
)  =  ( (
sum_ p  e.  (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  Prime ) ( log `  p
)  +  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) ( log `  p
) )  -  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
) ) )
72 fzfi 12051 . . . . . 6  |-  ( if ( M  <_  2 ,  M ,  2 ) ... M )  e. 
Fin
73 inss1 3718 . . . . . 6  |-  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)  i^i  Prime )  C_  ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)
74 ssfi 7741 . . . . . 6  |-  ( ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  e.  Fin  /\  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime )  C_  ( if ( M  <_  2 ,  M ,  2 ) ... M ) )  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime )  e.  Fin )
7572, 73, 74mp2an 672 . . . . 5  |-  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)  i^i  Prime )  e. 
Fin
7675a1i 11 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime )  e.  Fin )
77 ssun1 3667 . . . . . . 7  |-  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)  i^i  Prime )  C_  ( ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime )  u.  (
( ( M  + 
1 ) ... N
)  i^i  Prime ) )
7877, 44syl5sseqr 3553 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime )  C_  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... N
)  i^i  Prime ) )
7978sselda 3504 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) )  ->  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )
8079, 56syldan 470 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
8176, 80fsumcl 13521 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
)  e.  CC )
82 fzfi 12051 . . . . . 6  |-  ( ( M  +  1 ) ... N )  e. 
Fin
83 inss1 3718 . . . . . 6  |-  ( ( ( M  +  1 ) ... N )  i^i  Prime )  C_  (
( M  +  1 ) ... N )
84 ssfi 7741 . . . . . 6  |-  ( ( ( ( M  + 
1 ) ... N
)  e.  Fin  /\  ( ( ( M  +  1 ) ... N )  i^i  Prime ) 
C_  ( ( M  +  1 ) ... N ) )  -> 
( ( ( M  +  1 ) ... N )  i^i  Prime )  e.  Fin )
8582, 83, 84mp2an 672 . . . . 5  |-  ( ( ( M  +  1 ) ... N )  i^i  Prime )  e.  Fin
8685a1i 11 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
( M  +  1 ) ... N )  i^i  Prime )  e.  Fin )
87 ssun2 3668 . . . . . . 7  |-  ( ( ( M  +  1 ) ... N )  i^i  Prime )  C_  (
( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime )  u.  (
( ( M  + 
1 ) ... N
)  i^i  Prime ) )
8887, 44syl5sseqr 3553 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
( M  +  1 ) ... N )  i^i  Prime )  C_  (
( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i  Prime ) )
8988sselda 3504 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) )  ->  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )
9089, 56syldan 470 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) )  ->  ( log `  p )  e.  CC )
9186, 90fsumcl 13521 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) ( log `  p
)  e.  CC )
9281, 91pncan2d 9933 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( sum_ p  e.  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)  i^i  Prime ) ( log `  p )  +  sum_ p  e.  ( ( ( M  + 
1 ) ... N
)  i^i  Prime ) ( log `  p ) )  -  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
) )  =  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i 
Prime ) ( log `  p
) )
9371, 92eqtrd 2508 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( theta `  N )  -  ( theta `  M )
)  =  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) ( log `  p
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ifcif 3939   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Fincfn 7517   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    < clt 9629    <_ cle 9630    - cmin 9806   NNcn 10537   2c2 10586   ZZcz 10865   ZZ>=cuz 11083   [,]cicc 11533   ...cfz 11673   |_cfl 11896   sum_csu 13474   Primecprime 14079   logclog 22767   thetaccht 23189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-fi 7872  df-sup 7902  df-oi 7936  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-ioo 11534  df-ioc 11535  df-ico 11536  df-icc 11537  df-fz 11674  df-fzo 11794  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-fac 12323  df-bc 12350  df-hash 12375  df-shft 12866  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-limsup 13260  df-clim 13277  df-rlim 13278  df-sum 13475  df-ef 13668  df-sin 13670  df-cos 13671  df-pi 13673  df-dvds 13851  df-prm 14080  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-hom 14582  df-cco 14583  df-rest 14681  df-topn 14682  df-0g 14700  df-gsum 14701  df-topgen 14702  df-pt 14703  df-prds 14706  df-xrs 14760  df-qtop 14765  df-imas 14766  df-xps 14768  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-submnd 15790  df-mulg 15874  df-cntz 16169  df-cmn 16615  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-fbas 18227  df-fg 18228  df-cnfld 18232  df-top 19206  df-bases 19208  df-topon 19209  df-topsp 19210  df-cld 19326  df-ntr 19327  df-cls 19328  df-nei 19405  df-lp 19443  df-perf 19444  df-cn 19534  df-cnp 19535  df-haus 19622  df-tx 19890  df-hmeo 20083  df-fil 20174  df-fm 20266  df-flim 20267  df-flf 20268  df-xms 20650  df-ms 20651  df-tms 20652  df-cncf 21209  df-limc 22097  df-dv 22098  df-log 22769  df-cht 23195
This theorem is referenced by:  efchtdvds  23258
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