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Theorem prmorcht 23578
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 10563 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  RR )
2 chtval 23510 . . . . . . 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 2eluzge1 11152 . . . . . . . . . 10  |-  2  e.  ( ZZ>= `  1 )
5 ppisval2 23504 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ( ZZ>= ` 
1 ) )  -> 
( ( 0 [,] A )  i^i  Prime )  =  ( ( 1 ... ( |_ `  A ) )  i^i 
Prime ) )
61, 4, 5sylancl 662 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... ( |_ `  A
) )  i^i  Prime ) )
7 nnz 10907 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  ZZ )
8 flid 11948 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
97, 8syl 16 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( |_ `  A )  =  A )
109oveq2d 6312 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
1 ... ( |_ `  A ) )  =  ( 1 ... A
) )
1110ineq1d 3695 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 1 ... ( |_ `  A ) )  i^i  Prime )  =  ( ( 1 ... A
)  i^i  Prime ) )
126, 11eqtrd 2498 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... A )  i^i  Prime ) )
1312sumeq1d 13535 . . . . . . 7  |-  ( A  e.  NN  ->  sum_ k  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  k
)  =  sum_ k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
) )
14 inss1 3714 . . . . . . . 8  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
1514sseli 3495 . . . . . . . . . 10  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  ->  k  e.  ( 1 ... A
) )
16 elfznn 11739 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 1 ... A )  ->  k  e.  NN )
1716adantl 466 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  NN )
1817nnrpd 11280 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  RR+ )
1918relogcld 23134 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  RR )
2019recnd 9639 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  CC )
2115, 20sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  k  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  k )  e.  CC )
2221ralrimiva 2871 . . . . . . . 8  |-  ( A  e.  NN  ->  A. k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
)  e.  CC )
23 fzfi 12085 . . . . . . . . . 10  |-  ( 1 ... A )  e. 
Fin
2423olci 391 . . . . . . . . 9  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
25 sumss2 13560 . . . . . . . . 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 ) )
2624, 25mpan2 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 ) )
2714, 22, 26sylancr 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 ) )
2813, 27eqtrd 2498 . . . . . 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 ) )
293, 28eqtrd 2498 . . . . 5  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  k
) ,  0 ) )
30 elin 3683 . . . . . . . 8  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( k  e.  ( 1 ... A
)  /\  k  e.  Prime ) )
3130baibr 904 . . . . . . 7  |-  ( k  e.  ( 1 ... A )  ->  (
k  e.  Prime  <->  k  e.  ( ( 1 ... A )  i^i  Prime ) ) )
3231ifbid 3966 . . . . . 6  |-  ( k  e.  ( 1 ... A )  ->  if ( k  e.  Prime ,  ( log `  k
) ,  0 )  =  if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 ) )
3332sumeq2i 13533 . . . . 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 )
3429, 33syl6eqr 2516 . . . 4  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e. 
Prime ,  ( log `  k ) ,  0 ) )
35 eleq1 2529 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
36 fveq2 5872 . . . . . . . 8  |-  ( n  =  k  ->  ( log `  n )  =  ( log `  k
) )
3735, 36ifbieq1d 3967 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( log `  n
) ,  0 )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
38 eqid 2457 . . . . . . 7  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )
39 fvex 5882 . . . . . . . 8  |-  ( log `  k )  e.  _V
40 0cn 9605 . . . . . . . . 9  |-  0  e.  CC
4140elexi 3119 . . . . . . . 8  |-  0  e.  _V
4239, 41ifex 4013 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e. 
_V
4337, 38, 42fvmpt 5956 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) `  k )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
4417, 43syl 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 ) )
45 elnnuz 11142 . . . . . 6  |-  ( A  e.  NN  <->  A  e.  ( ZZ>= `  1 )
)
4645biimpi 194 . . . . 5  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  1 )
)
47 ifcl 3986 . . . . . 6  |-  ( ( ( log `  k
)  e.  CC  /\  0  e.  CC )  ->  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )  e.  CC )
4820, 40, 47sylancl 662 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e.  CC )
4944, 46, 48fsumser 13564 . . . 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
) )
5034, 49eqtrd 2498 . . 3  |-  ( A  e.  NN  ->  ( theta `  A )  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) ) `  A ) )
5150fveq2d 5876 . 2  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  ( exp `  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) ) )
52 addcl 9591 . . . 4  |-  ( ( k  e.  CC  /\  p  e.  CC )  ->  ( k  +  p
)  e.  CC )
5352adantl 466 . . 3  |-  ( ( A  e.  NN  /\  ( k  e.  CC  /\  p  e.  CC ) )  ->  ( k  +  p )  e.  CC )
5444, 48eqeltrd 2545 . . 3  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) `
 k )  e.  CC )
55 efadd 13841 . . . 4  |-  ( ( k  e.  CC  /\  p  e.  CC )  ->  ( exp `  (
k  +  p ) )  =  ( ( exp `  k )  x.  ( exp `  p
) ) )
5655adantl 466 . . 3  |-  ( ( A  e.  NN  /\  ( k  e.  CC  /\  p  e.  CC ) )  ->  ( exp `  ( k  +  p
) )  =  ( ( exp `  k
)  x.  ( exp `  p ) ) )
57 1nn 10567 . . . . . . 7  |-  1  e.  NN
58 ifcl 3986 . . . . . . 7  |-  ( ( k  e.  NN  /\  1  e.  NN )  ->  if ( k  e. 
Prime ,  k , 
1 )  e.  NN )
5917, 57, 58sylancl 662 . . . . . 6  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  NN )
6059nnrpd 11280 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  RR+ )
6160reeflogd 23135 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  ( log `  if ( k  e.  Prime ,  k ,  1 ) ) )  =  if ( k  e.  Prime ,  k ,  1 ) )
62 fvif 5883 . . . . . . 7  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  ( log `  1 ) )
63 log1 23096 . . . . . . . 8  |-  ( log `  1 )  =  0
64 ifeq2 3949 . . . . . . . 8  |-  ( ( log `  1 )  =  0  ->  if ( k  e.  Prime ,  ( log `  k
) ,  ( log `  1 ) )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
6563, 64ax-mp 5 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  ( log `  1
) )  =  if ( k  e.  Prime ,  ( log `  k
) ,  0 )
6662, 65eqtri 2486 . . . . . 6  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )
6744, 66syl6eqr 2516 . . . . 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 ) ) )
6867fveq2d 5876 . . . 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 ) ) ) )
69 id 22 . . . . . . 7  |-  ( n  =  k  ->  n  =  k )
7035, 69ifbieq1d 3967 . . . . . 6  |-  ( n  =  k  ->  if ( n  e.  Prime ,  n ,  1 )  =  if ( k  e.  Prime ,  k ,  1 ) )
71 prmorcht.1 . . . . . 6  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  n ,  1 ) )
72 vex 3112 . . . . . . 7  |-  k  e. 
_V
7357elexi 3119 . . . . . . 7  |-  1  e.  _V
7472, 73ifex 4013 . . . . . 6  |-  if ( k  e.  Prime ,  k ,  1 )  e. 
_V
7570, 71, 74fvmpt 5956 . . . . 5  |-  ( k  e.  NN  ->  ( F `  k )  =  if ( k  e. 
Prime ,  k , 
1 ) )
7617, 75syl 16 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( F `  k )  =  if ( k  e.  Prime ,  k ,  1 ) )
7761, 68, 763eqtr4d 2508 . . 3  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) `  k ) )  =  ( F `
 k ) )
7853, 54, 46, 56, 77seqhomo 12157 . 2  |-  ( A  e.  NN  ->  ( exp `  (  seq 1
(  +  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) )  =  (  seq 1 (  x.  ,  F ) `  A ) )
7951, 78eqtrd 2498 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 1395    e. wcel 1819   A.wral 2807    i^i cin 3470    C_ wss 3471   ifcif 3944    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   Fincfn 7535   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   NNcn 10556   2c2 10606   ZZcz 10885   ZZ>=cuz 11106   [,]cicc 11557   ...cfz 11697   |_cfl 11930    seqcseq 12110   sum_csu 13520   expce 13809   Primecprime 14229   logclog 23068   thetaccht 23490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-pi 13820  df-dvds 13999  df-prm 14230  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397  df-log 23070  df-cht 23496
This theorem is referenced by:  chtublem  23612  bposlem6  23690
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