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Theorem chpval2 22516
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 22419 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) (Λ `  n
) )
2 fveq2 5688 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
3 id 22 . . 3  |-  ( A  e.  RR  ->  A  e.  RR )
4 elfznn 11474 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
54adantl 463 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
6 vmacl 22415 . . . . 5  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
75, 6syl 16 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  RR )
87recnd 9408 . . 3  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  CC )
9 simprr 751 . . 3  |-  ( ( A  e.  RR  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  (Λ `  n )  =  0 ) )  -> 
(Λ `  n )  =  0 )
102, 3, 8, 9fsumvma2 22512 . 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 3568 . . . . . . 7  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
12 simpr 458 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
1311, 12sseldi 3351 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
14 elfznn 11474 . . . . . 6  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
15 vmappw 22413 . . . . . 6  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
1613, 14, 15syl2an 474 . . . . 5  |-  ( ( ( A  e.  RR  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
1716sumeq2dv 13176 . . . 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 11791 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
19 prmuz2 13777 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
20 eluzelre 10867 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
21 eluz2b2 10923 . . . . . . . . . 10  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
2221simprbi 461 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2320, 22rplogcld 22037 . . . . . . . 8  |-  ( p  e.  ( ZZ>= `  2
)  ->  ( log `  p )  e.  RR+ )
2413, 19, 233syl 20 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
2524rpcnd 11025 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
26 fsumconst 13253 . . . . . 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 ) ) )
2718, 25, 26syl2anc 656 . . . . 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 ) ) )
28 ppisval 22400 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
29 inss1 3567 . . . . . . . . . . . . . 14  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
3028, 29syl6eqss 3403 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( 2 ... ( |_ `  A ) ) )
3130sselda 3353 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 2 ... ( |_ `  A ) ) )
32 elfzuz2 11452 . . . . . . . . . . . 12  |-  ( p  e.  ( 2 ... ( |_ `  A
) )  ->  ( |_ `  A )  e.  ( ZZ>= `  2 )
)
3331, 32syl 16 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ( ZZ>= ` 
2 ) )
34 simpl 454 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  A  e.  RR )
35 0red 9383 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  e.  RR )
36 2re 10387 . . . . . . . . . . . . . 14  |-  2  e.  RR
3736a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  2  e.  RR )
38 2pos 10409 . . . . . . . . . . . . . 14  |-  0  <  2
3938a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  <  2 )
40 eluzle 10869 . . . . . . . . . . . . . . 15  |-  ( ( |_ `  A )  e.  ( ZZ>= `  2
)  ->  2  <_  ( |_ `  A ) )
41 2z 10674 . . . . . . . . . . . . . . . 16  |-  2  e.  ZZ
42 flge 11651 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  2  e.  ZZ )  ->  ( 2  <_  A  <->  2  <_  ( |_ `  A ) ) )
4341, 42mpan2 666 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  (
2  <_  A  <->  2  <_  ( |_ `  A ) ) )
4440, 43syl5ibr 221 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( |_ `  A
)  e.  ( ZZ>= ` 
2 )  ->  2  <_  A ) )
4544imp 429 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  2  <_  A )
4635, 37, 34, 39, 45ltletrd 9527 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  <  A )
4734, 46elrpd 11021 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  A  e.  RR+ )
4833, 47syldan 467 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR+ )
4948relogcld 22031 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
5049, 24rerpdivcld 11050 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
51 1red 9397 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  e.  RR )
52 1lt2 10484 . . . . . . . . . . . . . 14  |-  1  <  2
5352a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  <  2 )
5451, 37, 34, 53, 45ltletrd 9527 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  <  A )
5533, 54syldan 467 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
56 rplogcl 22012 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
5755, 56syldan 467 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
5857, 24rpdivcld 11040 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
5958rpge0d 11027 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
60 flge0nn0 11662 . . . . . . . 8  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
6150, 59, 60syl2anc 656 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
62 hashfz1 12113 . . . . . . 7  |-  ( ( |_ `  ( ( log `  A )  /  ( log `  p
) ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
6361, 62syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( # `  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
6463oveq1d 6105 . . . . 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 ) ) )
6561nn0cnd 10634 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
6665, 25mulcomd 9403 . . . . 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 ) ) ) ) )
6727, 64, 663eqtrd 2477 . . . 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 ) ) ) ) )
6817, 67eqtrd 2473 . . 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 ) ) ) ) )
6968sumeq2dv 13176 . 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 ) ) ) ) )
701, 10, 693eqtrd 2477 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    i^i cin 3324   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Fincfn 7306   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    < clt 9414    <_ cle 9415    / cdiv 9989   NNcn 10318   2c2 10367   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   RR+crp 10987   [,]cicc 11299   ...cfz 11433   |_cfl 11636   ^cexp 11861   #chash 12099   sum_csu 13159   Primecprime 13759   logclog 21965  Λcvma 22388  ψcchp 22389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-vma 22394  df-chp 22395
This theorem is referenced by:  chpchtsum  22517  chpub  22518
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