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Theorem chpval2 23764
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 23667 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) (Λ `  n
) )
2 fveq2 5803 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
3 id 22 . . 3  |-  ( A  e.  RR  ->  A  e.  RR )
4 elfznn 11683 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
54adantl 464 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
6 vmacl 23663 . . . . 5  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
75, 6syl 17 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  RR )
87recnd 9570 . . 3  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  CC )
9 simprr 756 . . 3  |-  ( ( A  e.  RR  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  (Λ `  n )  =  0 ) )  -> 
(Λ `  n )  =  0 )
102, 3, 8, 9fsumvma2 23760 . 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 3657 . . . . . . 7  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
12 simpr 459 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
1311, 12sseldi 3437 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
14 elfznn 11683 . . . . . 6  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
15 vmappw 23661 . . . . . 6  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
1613, 14, 15syl2an 475 . . . . 5  |-  ( ( ( A  e.  RR  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
1716sumeq2dv 13579 . . . 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 12035 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
19 prmuz2 14334 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
20 eluzelre 11053 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
21 eluz2b2 11115 . . . . . . . . . 10  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
2221simprbi 462 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2320, 22rplogcld 23198 . . . . . . . 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 11222 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
26 fsumconst 13661 . . . . . 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 659 . . . . 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 23648 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
29 inss1 3656 . . . . . . . . . . . . . 14  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
3028, 29syl6eqss 3489 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( 2 ... ( |_ `  A ) ) )
3130sselda 3439 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 2 ... ( |_ `  A ) ) )
32 elfzuz2 11660 . . . . . . . . . . . 12  |-  ( p  e.  ( 2 ... ( |_ `  A
) )  ->  ( |_ `  A )  e.  ( ZZ>= `  2 )
)
3331, 32syl 17 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ( ZZ>= ` 
2 ) )
34 simpl 455 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  A  e.  RR )
35 0red 9545 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  e.  RR )
36 2re 10564 . . . . . . . . . . . . . 14  |-  2  e.  RR
3736a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  2  e.  RR )
38 2pos 10586 . . . . . . . . . . . . . 14  |-  0  <  2
3938a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  <  2 )
40 eluzle 11055 . . . . . . . . . . . . . . 15  |-  ( ( |_ `  A )  e.  ( ZZ>= `  2
)  ->  2  <_  ( |_ `  A ) )
41 2z 10855 . . . . . . . . . . . . . . . 16  |-  2  e.  ZZ
42 flge 11890 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  2  e.  ZZ )  ->  ( 2  <_  A  <->  2  <_  ( |_ `  A ) ) )
4341, 42mpan2 669 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  (
2  <_  A  <->  2  <_  ( |_ `  A ) ) )
4440, 43syl5ibr 221 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
( |_ `  A
)  e.  ( ZZ>= ` 
2 )  ->  2  <_  A ) )
4544imp 427 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  2  <_  A )
4635, 37, 34, 39, 45ltletrd 9694 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  0  <  A )
4734, 46elrpd 11217 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  A  e.  RR+ )
4833, 47syldan 468 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR+ )
4948relogcld 23192 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
5049, 24rerpdivcld 11247 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
51 1red 9559 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  e.  RR )
52 1lt2 10661 . . . . . . . . . . . . . 14  |-  1  <  2
5352a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  <  2 )
5451, 37, 34, 53, 45ltletrd 9694 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  2
) )  ->  1  <  A )
5533, 54syldan 468 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
56 rplogcl 23173 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
5755, 56syldan 468 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
5857, 24rpdivcld 11237 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
5958rpge0d 11224 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
60 flge0nn0 11903 . . . . . . . 8  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
6150, 59, 60syl2anc 659 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
62 hashfz1 12371 . . . . . . 7  |-  ( ( |_ `  ( ( log `  A )  /  ( log `  p
) ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
6361, 62syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( # `  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
6463oveq1d 6247 . . . . 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 10813 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
6665, 25mulcomd 9565 . . . . 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 2445 . . . 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 2441 . . 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 13579 . 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 2445 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 367    = wceq 1403    e. wcel 1840    i^i cin 3410   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Fincfn 7472   CCcc 9438   RRcr 9439   0cc0 9440   1c1 9441    x. cmul 9445    < clt 9576    <_ cle 9577    / cdiv 10165   NNcn 10494   2c2 10544   NN0cn0 10754   ZZcz 10823   ZZ>=cuz 11043   RR+crp 11181   [,]cicc 11501   ...cfz 11641   |_cfl 11875   ^cexp 12118   #chash 12357   sum_csu 13562   Primecprime 14316   logclog 23124  Λcvma 23636  ψcchp 23637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-fi 7823  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-ioo 11502  df-ioc 11503  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-fl 11877  df-mod 11946  df-seq 12060  df-exp 12119  df-fac 12306  df-bc 12333  df-hash 12358  df-shft 12954  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-limsup 13348  df-clim 13365  df-rlim 13366  df-sum 13563  df-ef 13902  df-sin 13904  df-cos 13905  df-pi 13907  df-dvds 14086  df-gcd 14244  df-prm 14317  df-pc 14460  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-rest 14927  df-topn 14928  df-0g 14946  df-gsum 14947  df-topgen 14948  df-pt 14949  df-prds 14952  df-xrs 15006  df-qtop 15011  df-imas 15012  df-xps 15014  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-mulg 16274  df-cntz 16569  df-cmn 17014  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624  df-mopn 18625  df-fbas 18626  df-fg 18627  df-cnfld 18631  df-top 19581  df-bases 19583  df-topon 19584  df-topsp 19585  df-cld 19702  df-ntr 19703  df-cls 19704  df-nei 19782  df-lp 19820  df-perf 19821  df-cn 19911  df-cnp 19912  df-haus 19999  df-tx 20245  df-hmeo 20438  df-fil 20529  df-fm 20621  df-flim 20622  df-flf 20623  df-xms 21005  df-ms 21006  df-tms 21007  df-cncf 21564  df-limc 22452  df-dv 22453  df-log 23126  df-vma 23642  df-chp 23643
This theorem is referenced by:  chpchtsum  23765  chpub  23766
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