MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chpval2 Unicode version

Theorem chpval2 20963
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 20866 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) (Λ `  n
) )
2 fveq2 5695 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
3 id 20 . . 3  |-  ( A  e.  RR  ->  A  e.  RR )
4 elfznn 11044 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
54adantl 453 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
6 vmacl 20862 . . . . 5  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
75, 6syl 16 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  RR )
87recnd 9078 . . 3  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  CC )
9 simprr 734 . . 3  |-  ( ( A  e.  RR  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  (Λ `  n )  =  0 ) )  -> 
(Λ `  n )  =  0 )
102, 3, 8, 9fsumvma2 20959 . 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 3530 . . . . . . 7  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
12 simpr 448 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
1311, 12sseldi 3314 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
14 elfznn 11044 . . . . . 6  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
15 vmappw 20860 . . . . . 6  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
1613, 14, 15syl2an 464 . . . . 5  |-  ( ( ( A  e.  RR  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
1716sumeq2dv 12460 . . . 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 11275 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
19 prmuz2 13060 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
20 eluzelre 10461 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
21 eluz2b2 10512 . . . . . . . . . 10  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
2221simprbi 451 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2320, 22rplogcld 20485 . . . . . . . 8  |-  ( p  e.  ( ZZ>= `  2
)  ->  ( log `  p )  e.  RR+ )
2413, 19, 233syl 19 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
2524rpcnd 10614 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
26 fsumconst 12536 . . . . . 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 643 . . . . 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 20847 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
29 inss1 3529 . . . . . . . . . . . . . 14  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
3028, 29syl6eqss 3366 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( 2 ... ( |_ `  A ) ) )
3130sselda 3316 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 2 ... ( |_ `  A ) ) )
32 elfzuz2 11026 . . . . . . . . . . . 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 444 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  A  e.  RR )
35 0re 9055 . . . . . . . . . . . . . 14  |-  0  e.  RR
3635a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  e.  RR )
37 2re 10033 . . . . . . . . . . . . . 14  |-  2  e.  RR
3837a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  2  e.  RR )
39 2pos 10046 . . . . . . . . . . . . . 14  |-  0  <  2
4039a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  <  2 )
41 eluzle 10462 . . . . . . . . . . . . . . 15  |-  ( ( |_ `  A )  e.  ( ZZ>= `  2
)  ->  2  <_  ( |_ `  A ) )
42 2z 10276 . . . . . . . . . . . . . . . 16  |-  2  e.  ZZ
43 flge 11177 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  2  e.  ZZ )  ->  ( 2  <_  A  <->  2  <_  ( |_ `  A ) ) )
4442, 43mpan2 653 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  (
2  <_  A  <->  2  <_  ( |_ `  A ) ) )
4541, 44syl5ibr 213 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( |_ `  A
)  e.  ( ZZ>= ` 
2 )  ->  2  <_  A ) )
4645imp 419 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  2  <_  A )
4736, 38, 34, 40, 46ltletrd 9194 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  <  A )
4834, 47elrpd 10610 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  A  e.  RR+ )
4933, 48syldan 457 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR+ )
5049relogcld 20479 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
5150, 24rerpdivcld 10639 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
52 1re 9054 . . . . . . . . . . . . . 14  |-  1  e.  RR
5352a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  e.  RR )
54 1lt2 10106 . . . . . . . . . . . . . 14  |-  1  <  2
5554a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  <  2 )
5653, 38, 34, 55, 46ltletrd 9194 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  <  A )
5733, 56syldan 457 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
58 rplogcl 20460 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
5957, 58syldan 457 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
6059, 24rpdivcld 10629 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
6160rpge0d 10616 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
62 flge0nn0 11188 . . . . . . . 8  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
6351, 61, 62syl2anc 643 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
64 hashfz1 11593 . . . . . . 7  |-  ( ( |_ `  ( ( log `  A )  /  ( log `  p
) ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
6563, 64syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( # `  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
6665oveq1d 6063 . . . . 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 ) ) )
6763nn0cnd 10240 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
6867, 25mulcomd 9073 . . . . 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 ) ) ) ) )
6927, 66, 683eqtrd 2448 . . . 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 ) ) ) ) )
7017, 69eqtrd 2444 . . 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 ) ) ) ) )
7170sumeq2dv 12460 . 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 ) ) ) ) )
721, 10, 713eqtrd 2448 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3287   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Fincfn 7076   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955    x. cmul 8959    < clt 9084    <_ cle 9085    / cdiv 9641   NNcn 9964   2c2 10013   NN0cn0 10185   ZZcz 10246   ZZ>=cuz 10452   RR+crp 10576   [,]cicc 10883   ...cfz 11007   |_cfl 11164   ^cexp 11345   #chash 11581   sum_csu 12442   Primecprime 13042   logclog 20413  Λcvma 20835  ψcchp 20836
This theorem is referenced by:  chpchtsum  20964  chpub  20965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635  df-cos 12636  df-pi 12638  df-dvds 12816  df-gcd 12970  df-prm 13043  df-pc 13174  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715  df-log 20415  df-vma 20841  df-chp 20842
  Copyright terms: Public domain W3C validator