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Theorem chtdif 24164
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 11193 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  RR )
2 chtval 24116 . . . . 5  |-  ( N  e.  RR  ->  ( theta `  N )  = 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p ) )
31, 2syl 17 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( theta `  N )  =  sum_ p  e.  ( ( 0 [,] N )  i^i 
Prime ) ( log `  p
) )
4 eluzel2 11187 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5 2z 10993 . . . . . . . . . 10  |-  2  e.  ZZ
6 ifcl 3914 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  2  e.  ZZ )  ->  if ( M  <_ 
2 ,  M , 
2 )  e.  ZZ )
74, 5, 6sylancl 675 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  if ( M  <_  2 ,  M ,  2 )  e.  ZZ )
85a1i 11 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  2  e.  ZZ )
94zred 11063 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  RR )
10 2re 10701 . . . . . . . . . 10  |-  2  e.  RR
11 min2 11507 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  2  e.  RR )  ->  if ( M  <_ 
2 ,  M , 
2 )  <_  2
)
129, 10, 11sylancl 675 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  if ( M  <_  2 ,  M ,  2 )  <_ 
2 )
13 eluz2 11188 . . . . . . . . 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 1214 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  2  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) ) )
15 ppisval2 24110 . . . . . . . 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 673 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
0 [,] N )  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  N
) )  i^i  Prime ) )
17 eluzelz 11192 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
18 flid 12077 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( |_ `  N )  =  N )
1917, 18syl 17 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( |_ `  N )  =  N )
2019oveq2d 6324 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  N ) )  =  ( if ( M  <_  2 ,  M ,  2 ) ... N ) )
2120ineq1d 3624 . . . . . . 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 2505 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
0 [,] N )  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i  Prime ) )
2322sumeq1d 13844 . . . . 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 10559 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <  ( M  +  1 ) )
25 fzdisj 11852 . . . . . . . . 9  |-  ( M  <  ( M  + 
1 )  ->  (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  (
( M  +  1 ) ... N ) )  =  (/) )
2624, 25syl 17 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  ( ( M  + 
1 ) ... N
) )  =  (/) )
2726ineq1d 3624 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  (
( M  +  1 ) ... N ) )  i^i  Prime )  =  ( (/)  i^i  Prime ) )
28 inindir 3641 . . . . . . 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 3616 . . . . . . . 8  |-  ( (/)  i^i 
Prime )  =  ( Prime  i^i  (/) )
30 in0 3763 . . . . . . . 8  |-  ( Prime  i^i  (/) )  =  (/)
3129, 30eqtri 2493 . . . . . . 7  |-  ( (/)  i^i 
Prime )  =  (/)
3227, 28, 313eqtr3g 2528 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  Prime )  i^i  ( ( ( M  +  1 ) ... N )  i^i 
Prime ) )  =  (/) )
33 min1 11506 . . . . . . . . . . . 12  |-  ( ( M  e.  RR  /\  2  e.  RR )  ->  if ( M  <_ 
2 ,  M , 
2 )  <_  M
)
349, 10, 33sylancl 675 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  M
)  ->  if ( M  <_  2 ,  M ,  2 )  <_  M )
35 eluz2 11188 . . . . . . . . . . 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 1214 . . . . . . . . . 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 11820 . . . . . . . . . 10  |-  ( M  e.  ( if ( M  <_  2 ,  M ,  2 ) ... N )  <->  ( M  e.  ( ZZ>= `  if ( M  <_  2 ,  M ,  2 ) )  /\  N  e.  (
ZZ>= `  M ) ) )
3936, 37, 38sylanbrc 677 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( if ( M  <_ 
2 ,  M , 
2 ) ... N
) )
40 fzsplit 11851 . . . . . . . . 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 17 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( if ( M  <_  2 ,  M ,  2 ) ... N )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  u.  ( ( M  + 
1 ) ... N
) ) )
4241ineq1d 3624 . . . . . . 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 3682 . . . . . . 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 2521 . . . . . 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 12224 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( if ( M  <_  2 ,  M ,  2 ) ... N )  e. 
Fin )
46 inss1 3643 . . . . . . 7  |-  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... N
)  i^i  Prime )  C_  ( if ( M  <_ 
2 ,  M , 
2 ) ... N
)
47 ssfi 7810 . . . . . . 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 675 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime )  e.  Fin )
49 inss2 3644 . . . . . . . . . . 11  |-  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... N
)  i^i  Prime )  C_  Prime
50 simpr 468 . . . . . . . . . . 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 3416 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  p  e.  Prime )
52 prmnn 14704 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
5351, 52syl 17 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  p  e.  NN )
5453nnrpd 11362 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  p  e.  RR+ )
5554relogcld 23651 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
5655recnd 9687 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
5732, 44, 48, 56fsumsplit 13883 . . . . 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 2505 . . . 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 2505 . . 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 24116 . . . . 5  |-  ( M  e.  RR  ->  ( theta `  M )  = 
sum_ p  e.  (
( 0 [,] M
)  i^i  Prime ) ( log `  p ) )
619, 60syl 17 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( theta `  M )  =  sum_ p  e.  ( ( 0 [,] M )  i^i 
Prime ) ( log `  p
) )
62 ppisval2 24110 . . . . . . 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 673 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
0 [,] M )  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  M
) )  i^i  Prime ) )
64 flid 12077 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( |_ `  M )  =  M )
654, 64syl 17 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( |_ `  M )  =  M )
6665oveq2d 6324 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( if ( M  <_  2 ,  M ,  2 ) ... ( |_ `  M ) )  =  ( if ( M  <_  2 ,  M ,  2 ) ... M ) )
6766ineq1d 3624 . . . . . 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 2505 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
0 [,] M )  i^i  Prime )  =  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i  Prime ) )
6968sumeq1d 13844 . . . 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 2505 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( theta `  M )  =  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
) )
7159, 70oveq12d 6326 . 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 12223 . . . . . 6  |-  ( if ( M  <_  2 ,  M ,  2 ) ... M )  e. 
Fin
73 inss1 3643 . . . . . 6  |-  ( ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)  i^i  Prime )  C_  ( if ( M  <_ 
2 ,  M , 
2 ) ... M
)
74 ssfi 7810 . . . . . 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 686 . . . . 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 3588 . . . . . . 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 3467 . . . . . 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 3418 . . . . 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 478 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
8176, 80fsumcl 13876 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  sum_ p  e.  ( ( if ( M  <_  2 ,  M ,  2 ) ... M )  i^i 
Prime ) ( log `  p
)  e.  CC )
82 fzfi 12223 . . . . . 6  |-  ( ( M  +  1 ) ... N )  e. 
Fin
83 inss1 3643 . . . . . 6  |-  ( ( ( M  +  1 ) ... N )  i^i  Prime )  C_  (
( M  +  1 ) ... N )
84 ssfi 7810 . . . . . 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 686 . . . . 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 3589 . . . . . . 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 3467 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
( M  +  1 ) ... N )  i^i  Prime )  C_  (
( if ( M  <_  2 ,  M ,  2 ) ... N )  i^i  Prime ) )
8988sselda 3418 . . . . 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 478 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) )  ->  ( log `  p )  e.  CC )
9186, 90fsumcl 13876 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  sum_ p  e.  ( ( ( M  +  1 ) ... N )  i^i  Prime ) ( log `  p
)  e.  CC )
9281, 91pncan2d 10007 . 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 2505 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 376    = wceq 1452    e. wcel 1904    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   ifcif 3872   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   2c2 10681   ZZcz 10961   ZZ>=cuz 11182   [,]cicc 11663   ...cfz 11810   |_cfl 12059   sum_csu 13829   Primecprime 14701   logclog 23583   thetaccht 24096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-prm 14702  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cht 24102
This theorem is referenced by:  efchtdvds  24165
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