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Theorem prmorcht 22491
Description: Relate the primorial (product of the first  n primes) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)
Hypothesis
Ref Expression
prmorcht.1  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  n ,  1 ) )
Assertion
Ref Expression
prmorcht  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  (  seq 1 (  x.  ,  F ) `  A ) )

Proof of Theorem prmorcht
Dummy variables  k  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnre 10321 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  RR )
2 chtval 22423 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ k  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  k ) )
31, 2syl 16 . . . . . 6  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  k ) )
4 2nn 10471 . . . . . . . . . . 11  |-  2  e.  NN
5 nnuz 10888 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
64, 5eleqtri 2510 . . . . . . . . . 10  |-  2  e.  ( ZZ>= `  1 )
7 ppisval2 22417 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ( ZZ>= ` 
1 ) )  -> 
( ( 0 [,] A )  i^i  Prime )  =  ( ( 1 ... ( |_ `  A ) )  i^i 
Prime ) )
81, 6, 7sylancl 662 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... ( |_ `  A
) )  i^i  Prime ) )
9 nnz 10660 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  ZZ )
10 flid 11649 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
119, 10syl 16 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( |_ `  A )  =  A )
1211oveq2d 6102 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
1 ... ( |_ `  A ) )  =  ( 1 ... A
) )
1312ineq1d 3546 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 1 ... ( |_ `  A ) )  i^i  Prime )  =  ( ( 1 ... A
)  i^i  Prime ) )
148, 13eqtrd 2470 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... A )  i^i  Prime ) )
1514sumeq1d 13170 . . . . . . 7  |-  ( A  e.  NN  ->  sum_ k  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  k
)  =  sum_ k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
) )
16 inss1 3565 . . . . . . . 8  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
1716sseli 3347 . . . . . . . . . 10  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  ->  k  e.  ( 1 ... A
) )
18 elfznn 11470 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 1 ... A )  ->  k  e.  NN )
1918adantl 466 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  NN )
2019nnrpd 11018 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  RR+ )
2120relogcld 22047 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  RR )
2221recnd 9404 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  CC )
2317, 22sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  k  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  k )  e.  CC )
2423ralrimiva 2794 . . . . . . . 8  |-  ( A  e.  NN  ->  A. k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
)  e.  CC )
25 fzfi 11786 . . . . . . . . . 10  |-  ( 1 ... A )  e. 
Fin
2625olci 391 . . . . . . . . 9  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
27 sumss2 13195 . . . . . . . . 9  |-  ( ( ( ( ( 1 ... A )  i^i 
Prime )  C_  ( 1 ... A )  /\  A. k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k )  e.  CC )  /\  ( ( 1 ... A )  C_  ( ZZ>= `  1 )  \/  ( 1 ... A
)  e.  Fin )
)  ->  sum_ k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
)  =  sum_ k  e.  ( 1 ... A
) if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 ) )
2826, 27mpan2 671 . . . . . . . 8  |-  ( ( ( ( 1 ... A )  i^i  Prime ) 
C_  ( 1 ... A )  /\  A. k  e.  ( (
1 ... A )  i^i 
Prime ) ( log `  k
)  e.  CC )  ->  sum_ k  e.  ( ( 1 ... A
)  i^i  Prime ) ( log `  k )  =  sum_ k  e.  ( 1 ... A ) if ( k  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  k
) ,  0 ) )
2916, 24, 28sylancr 663 . . . . . . 7  |-  ( A  e.  NN  ->  sum_ k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
)  =  sum_ k  e.  ( 1 ... A
) if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 ) )
3015, 29eqtrd 2470 . . . . . 6  |-  ( A  e.  NN  ->  sum_ k  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  k
)  =  sum_ k  e.  ( 1 ... A
) if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 ) )
313, 30eqtrd 2470 . . . . 5  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  k
) ,  0 ) )
32 elin 3534 . . . . . . . 8  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( k  e.  ( 1 ... A
)  /\  k  e.  Prime ) )
3332baibr 897 . . . . . . 7  |-  ( k  e.  ( 1 ... A )  ->  (
k  e.  Prime  <->  k  e.  ( ( 1 ... A )  i^i  Prime ) ) )
3433ifbid 3806 . . . . . 6  |-  ( k  e.  ( 1 ... A )  ->  if ( k  e.  Prime ,  ( log `  k
) ,  0 )  =  if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 ) )
3534sumeq2i 13168 . . . . 5  |-  sum_ k  e.  ( 1 ... A
) if ( k  e.  Prime ,  ( log `  k ) ,  0 )  =  sum_ k  e.  ( 1 ... A
) if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 )
3631, 35syl6eqr 2488 . . . 4  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e. 
Prime ,  ( log `  k ) ,  0 ) )
37 eleq1 2498 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
38 fveq2 5686 . . . . . . . 8  |-  ( n  =  k  ->  ( log `  n )  =  ( log `  k
) )
3937, 38ifbieq1d 3807 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( log `  n
) ,  0 )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
40 eqid 2438 . . . . . . 7  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )
41 fvex 5696 . . . . . . . 8  |-  ( log `  k )  e.  _V
42 0cn 9370 . . . . . . . . 9  |-  0  e.  CC
4342elexi 2977 . . . . . . . 8  |-  0  e.  _V
4441, 43ifex 3853 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e. 
_V
4539, 40, 44fvmpt 5769 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) `  k )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
4619, 45syl 16 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) `
 k )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  0 ) )
47 elnnuz 10889 . . . . . 6  |-  ( A  e.  NN  <->  A  e.  ( ZZ>= `  1 )
)
4847biimpi 194 . . . . 5  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  1 )
)
49 ifcl 3826 . . . . . 6  |-  ( ( ( log `  k
)  e.  CC  /\  0  e.  CC )  ->  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )  e.  CC )
5022, 42, 49sylancl 662 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e.  CC )
5146, 48, 50fsumser 13199 . . . 4  |-  ( A  e.  NN  ->  sum_ k  e.  ( 1 ... A
) if ( k  e.  Prime ,  ( log `  k ) ,  0 )  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) )
5236, 51eqtrd 2470 . . 3  |-  ( A  e.  NN  ->  ( theta `  A )  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) ) `  A ) )
5352fveq2d 5690 . 2  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  ( exp `  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) ) )
54 addcl 9356 . . . 4  |-  ( ( k  e.  CC  /\  p  e.  CC )  ->  ( k  +  p
)  e.  CC )
5554adantl 466 . . 3  |-  ( ( A  e.  NN  /\  ( k  e.  CC  /\  p  e.  CC ) )  ->  ( k  +  p )  e.  CC )
5646, 50eqeltrd 2512 . . 3  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) `
 k )  e.  CC )
57 efadd 13371 . . . 4  |-  ( ( k  e.  CC  /\  p  e.  CC )  ->  ( exp `  (
k  +  p ) )  =  ( ( exp `  k )  x.  ( exp `  p
) ) )
5857adantl 466 . . 3  |-  ( ( A  e.  NN  /\  ( k  e.  CC  /\  p  e.  CC ) )  ->  ( exp `  ( k  +  p
) )  =  ( ( exp `  k
)  x.  ( exp `  p ) ) )
59 1nn 10325 . . . . . . 7  |-  1  e.  NN
60 ifcl 3826 . . . . . . 7  |-  ( ( k  e.  NN  /\  1  e.  NN )  ->  if ( k  e. 
Prime ,  k , 
1 )  e.  NN )
6119, 59, 60sylancl 662 . . . . . 6  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  NN )
6261nnrpd 11018 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  RR+ )
6362reeflogd 22048 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  ( log `  if ( k  e.  Prime ,  k ,  1 ) ) )  =  if ( k  e.  Prime ,  k ,  1 ) )
64 fvif 5697 . . . . . . 7  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  ( log `  1 ) )
65 log1 22009 . . . . . . . 8  |-  ( log `  1 )  =  0
66 ifeq2 3791 . . . . . . . 8  |-  ( ( log `  1 )  =  0  ->  if ( k  e.  Prime ,  ( log `  k
) ,  ( log `  1 ) )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
6765, 66ax-mp 5 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  ( log `  1
) )  =  if ( k  e.  Prime ,  ( log `  k
) ,  0 )
6864, 67eqtri 2458 . . . . . 6  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )
6946, 68syl6eqr 2488 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) `
 k )  =  ( log `  if ( k  e.  Prime ,  k ,  1 ) ) )
7069fveq2d 5690 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) `  k ) )  =  ( exp `  ( log `  if ( k  e.  Prime ,  k ,  1 ) ) ) )
71 id 22 . . . . . . 7  |-  ( n  =  k  ->  n  =  k )
7237, 71ifbieq1d 3807 . . . . . 6  |-  ( n  =  k  ->  if ( n  e.  Prime ,  n ,  1 )  =  if ( k  e.  Prime ,  k ,  1 ) )
73 prmorcht.1 . . . . . 6  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  n ,  1 ) )
74 vex 2970 . . . . . . 7  |-  k  e. 
_V
7559elexi 2977 . . . . . . 7  |-  1  e.  _V
7674, 75ifex 3853 . . . . . 6  |-  if ( k  e.  Prime ,  k ,  1 )  e. 
_V
7772, 73, 76fvmpt 5769 . . . . 5  |-  ( k  e.  NN  ->  ( F `  k )  =  if ( k  e. 
Prime ,  k , 
1 ) )
7819, 77syl 16 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( F `  k )  =  if ( k  e.  Prime ,  k ,  1 ) )
7963, 70, 783eqtr4d 2480 . . 3  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) `  k ) )  =  ( F `
 k ) )
8055, 56, 48, 58, 79seqhomo 11845 . 2  |-  ( A  e.  NN  ->  ( exp `  (  seq 1
(  +  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) )  =  (  seq 1 (  x.  ,  F ) `  A ) )
8153, 80eqtrd 2470 1  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  (  seq 1 (  x.  ,  F ) `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710    i^i cin 3322    C_ wss 3323   ifcif 3786    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   NNcn 10314   2c2 10363   ZZcz 10638   ZZ>=cuz 10853   [,]cicc 11295   ...cfz 11429   |_cfl 11632    seqcseq 11798   sum_csu 13155   expce 13339   Primecprime 13755   logclog 21981   thetaccht 22403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-prm 13756  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-cht 22409
This theorem is referenced by:  chtublem  22525  bposlem6  22603
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