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Theorem chpval2 20569
Description: Express the second Chebyshev function directly as a sum over the primes less than  A (instead of indirectly through the von Mangoldt function). (Contributed by Mario Carneiro, 8-Apr-2016.)
Assertion
Ref Expression
chpval2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
Distinct variable group:    A, p

Proof of Theorem chpval2
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 chpval 20472 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) (Λ `  n
) )
2 fveq2 5608 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
3 id 19 . . 3  |-  ( A  e.  RR  ->  A  e.  RR )
4 elfznn 10911 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
54adantl 452 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
6 vmacl 20468 . . . . 5  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
75, 6syl 15 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  RR )
87recnd 8951 . . 3  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  CC )
9 simprr 733 . . 3  |-  ( ( A  e.  RR  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  (Λ `  n )  =  0 ) )  -> 
(Λ `  n )  =  0 )
102, 3, 8, 9fsumvma2 20565 . 2  |-  ( A  e.  RR  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) ) (Λ `  n )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) (Λ `  (
p ^ k ) ) )
11 inss2 3466 . . . . . . 7  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
12 simpr 447 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
1311, 12sseldi 3254 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
14 elfznn 10911 . . . . . 6  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
15 vmappw 20466 . . . . . 6  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
1613, 14, 15syl2an 463 . . . . 5  |-  ( ( ( A  e.  RR  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
1716sumeq2dv 12273 . . . 4  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) (Λ `  (
p ^ k ) )  =  sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) ( log `  p ) )
18 fzfid 11127 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
19 prmuz2 12873 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
20 eluzelre 10331 . . . . . . . . . 10  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
21 eluz2b2 10382 . . . . . . . . . . 11  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
2221simprbi 450 . . . . . . . . . 10  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2320, 22rplogcld 20091 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  ( log `  p )  e.  RR+ )
2419, 23syl 15 . . . . . . . 8  |-  ( p  e.  Prime  ->  ( log `  p )  e.  RR+ )
2513, 24syl 15 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
2625rpcnd 10484 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
27 fsumconst 12349 . . . . . 6  |-  ( ( ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin  /\  ( log `  p
)  e.  CC )  ->  sum_ k  e.  ( 1 ... ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) ( log `  p )  =  ( ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  x.  ( log `  p ) ) )
2818, 26, 27syl2anc 642 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ( log `  p
)  =  ( (
# `  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) ) )  x.  ( log `  p ) ) )
29 ppisval 20453 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
30 inss1 3465 . . . . . . . . . . . . . . 15  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
3130a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) ) )
3229, 31eqsstrd 3288 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( 2 ... ( |_ `  A ) ) )
3332sselda 3256 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 2 ... ( |_ `  A ) ) )
34 elfzuz2 10893 . . . . . . . . . . . 12  |-  ( p  e.  ( 2 ... ( |_ `  A
) )  ->  ( |_ `  A )  e.  ( ZZ>= `  2 )
)
3533, 34syl 15 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ( ZZ>= ` 
2 ) )
36 simpl 443 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  A  e.  RR )
37 0re 8928 . . . . . . . . . . . . . 14  |-  0  e.  RR
3837a1i 10 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  e.  RR )
39 2re 9905 . . . . . . . . . . . . . 14  |-  2  e.  RR
4039a1i 10 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  2  e.  RR )
41 2pos 9918 . . . . . . . . . . . . . 14  |-  0  <  2
4241a1i 10 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  <  2 )
43 eluzle 10332 . . . . . . . . . . . . . . 15  |-  ( ( |_ `  A )  e.  ( ZZ>= `  2
)  ->  2  <_  ( |_ `  A ) )
44 2z 10146 . . . . . . . . . . . . . . . 16  |-  2  e.  ZZ
45 flge 11029 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  2  e.  ZZ )  ->  ( 2  <_  A  <->  2  <_  ( |_ `  A ) ) )
4644, 45mpan2 652 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  (
2  <_  A  <->  2  <_  ( |_ `  A ) ) )
4743, 46syl5ibr 212 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( |_ `  A
)  e.  ( ZZ>= ` 
2 )  ->  2  <_  A ) )
4847imp 418 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  2  <_  A )
4938, 40, 36, 42, 48ltletrd 9066 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  <  A )
5036, 49elrpd 10480 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  A  e.  RR+ )
5135, 50syldan 456 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR+ )
5251relogcld 20085 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
5352, 25rerpdivcld 10509 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
54 1re 8927 . . . . . . . . . . . . . 14  |-  1  e.  RR
5554a1i 10 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  e.  RR )
56 1lt2 9978 . . . . . . . . . . . . . 14  |-  1  <  2
5756a1i 10 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  <  2 )
5855, 40, 36, 57, 48ltletrd 9066 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  <  A )
5935, 58syldan 456 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
60 rplogcl 20066 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
6159, 60syldan 456 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
6261, 25rpdivcld 10499 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
6362rpge0d 10486 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
64 flge0nn0 11040 . . . . . . . 8  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
6553, 63, 64syl2anc 642 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
66 hashfz1 11438 . . . . . . 7  |-  ( ( |_ `  ( ( log `  A )  /  ( log `  p
) ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
6765, 66syl 15 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( # `  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
6867oveq1d 5960 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  x.  ( log `  p ) )  =  ( ( |_
`  ( ( log `  A )  /  ( log `  p ) ) )  x.  ( log `  p ) ) )
6965nn0cnd 10112 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
7069, 26mulcomd 8946 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  x.  ( log `  p
) )  =  ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) )
7128, 68, 703eqtrd 2394 . . . 4  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ( log `  p
)  =  ( ( log `  p )  x.  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )
7217, 71eqtrd 2390 . . 3  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) (Λ `  (
p ^ k ) )  =  ( ( log `  p )  x.  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )
7372sumeq2dv 12273 . 2  |-  ( A  e.  RR  ->  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime )
sum_ k  e.  ( 1 ... ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) (Λ `  (
p ^ k ) )  =  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) )
741, 10, 733eqtrd 2394 1  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Fincfn 6951   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832    < clt 8957    <_ cle 8958    / cdiv 9513   NNcn 9836   2c2 9885   NN0cn0 10057   ZZcz 10116   ZZ>=cuz 10322   RR+crp 10446   [,]cicc 10751   ...cfz 10874   |_cfl 11016   ^cexp 11197   #chash 11430   sum_csu 12255   Primecprime 12855   logclog 20019  Λcvma 20441  ψcchp 20442
This theorem is referenced by:  chpchtsum  20570  chpub  20571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ioc 10753  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-sum 12256  df-ef 12446  df-sin 12448  df-cos 12449  df-pi 12451  df-dvds 12629  df-gcd 12783  df-prm 12856  df-pc 12987  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cn 17063  df-cnp 17064  df-haus 17149  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cncf 18485  df-limc 19320  df-dv 19321  df-log 20021  df-vma 20447  df-chp 20448
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