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Theorem prmorcht 20432
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 9769 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  RR )
2 chtval 20364 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ k  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  k ) )
31, 2syl 15 . . . . . 6  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  k ) )
4 2nn 9893 . . . . . . . . . . 11  |-  2  e.  NN
5 nnuz 10279 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
64, 5eleqtri 2368 . . . . . . . . . 10  |-  2  e.  ( ZZ>= `  1 )
7 ppisval2 20358 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ( ZZ>= ` 
1 ) )  -> 
( ( 0 [,] A )  i^i  Prime )  =  ( ( 1 ... ( |_ `  A ) )  i^i 
Prime ) )
81, 6, 7sylancl 643 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... ( |_ `  A
) )  i^i  Prime ) )
9 nnz 10061 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  ZZ )
10 flid 10955 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
119, 10syl 15 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( |_ `  A )  =  A )
1211oveq2d 5890 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
1 ... ( |_ `  A ) )  =  ( 1 ... A
) )
1312ineq1d 3382 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 1 ... ( |_ `  A ) )  i^i  Prime )  =  ( ( 1 ... A
)  i^i  Prime ) )
148, 13eqtrd 2328 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... A )  i^i  Prime ) )
1514sumeq1d 12190 . . . . . . 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 3402 . . . . . . . 8  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
1716sseli 3189 . . . . . . . . . 10  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  ->  k  e.  ( 1 ... A
) )
18 elfznn 10835 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 1 ... A )  ->  k  e.  NN )
1918adantl 452 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  NN )
2019nnrpd 10405 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  RR+ )
2120relogcld 19990 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  RR )
2221recnd 8877 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  CC )
2317, 22sylan2 460 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  k  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  k )  e.  CC )
2423ralrimiva 2639 . . . . . . . 8  |-  ( A  e.  NN  ->  A. k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
)  e.  CC )
25 fzfi 11050 . . . . . . . . . 10  |-  ( 1 ... A )  e. 
Fin
2625olci 380 . . . . . . . . 9  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
27 sumss2 12215 . . . . . . . . 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 652 . . . . . . . 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 644 . . . . . . 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 2328 . . . . . 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 2328 . . . . 5  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  k
) ,  0 ) )
32 elin 3371 . . . . . . . 8  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( k  e.  ( 1 ... A
)  /\  k  e.  Prime ) )
3332baibr 872 . . . . . . 7  |-  ( k  e.  ( 1 ... A )  ->  (
k  e.  Prime  <->  k  e.  ( ( 1 ... A )  i^i  Prime ) ) )
3433ifbid 3596 . . . . . 6  |-  ( k  e.  ( 1 ... A )  ->  if ( k  e.  Prime ,  ( log `  k
) ,  0 )  =  if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 ) )
3534sumeq2i 12188 . . . . 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 2346 . . . 4  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e. 
Prime ,  ( log `  k ) ,  0 ) )
37 eleq1 2356 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
38 fveq2 5541 . . . . . . . 8  |-  ( n  =  k  ->  ( log `  n )  =  ( log `  k
) )
39 eqidd 2297 . . . . . . . 8  |-  ( n  =  k  ->  0  =  0 )
4037, 38, 39ifbieq12d 3600 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( log `  n
) ,  0 )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
41 eqid 2296 . . . . . . 7  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )
42 fvex 5555 . . . . . . . 8  |-  ( log `  k )  e.  _V
43 0cn 8847 . . . . . . . . 9  |-  0  e.  CC
4443elexi 2810 . . . . . . . 8  |-  0  e.  _V
4542, 44ifex 3636 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e. 
_V
4640, 41, 45fvmpt 5618 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) `  k )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
4719, 46syl 15 . . . . 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 ) )
48 elnnuz 10280 . . . . . 6  |-  ( A  e.  NN  <->  A  e.  ( ZZ>= `  1 )
)
4948biimpi 186 . . . . 5  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  1 )
)
50 ifcl 3614 . . . . . 6  |-  ( ( ( log `  k
)  e.  CC  /\  0  e.  CC )  ->  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )  e.  CC )
5122, 43, 50sylancl 643 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e.  CC )
5247, 49, 51fsumser 12219 . . . 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
) )
5336, 52eqtrd 2328 . . 3  |-  ( A  e.  NN  ->  ( theta `  A )  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) ) `  A ) )
5453fveq2d 5545 . 2  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  ( exp `  (  seq  1 (  +  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) ) )
55 addcl 8835 . . . 4  |-  ( ( k  e.  CC  /\  p  e.  CC )  ->  ( k  +  p
)  e.  CC )
5655adantl 452 . . 3  |-  ( ( A  e.  NN  /\  ( k  e.  CC  /\  p  e.  CC ) )  ->  ( k  +  p )  e.  CC )
5747, 51eqeltrd 2370 . . 3  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) `
 k )  e.  CC )
58 efadd 12391 . . . 4  |-  ( ( k  e.  CC  /\  p  e.  CC )  ->  ( exp `  (
k  +  p ) )  =  ( ( exp `  k )  x.  ( exp `  p
) ) )
5958adantl 452 . . 3  |-  ( ( A  e.  NN  /\  ( k  e.  CC  /\  p  e.  CC ) )  ->  ( exp `  ( k  +  p
) )  =  ( ( exp `  k
)  x.  ( exp `  p ) ) )
60 1nn 9773 . . . . . . 7  |-  1  e.  NN
61 ifcl 3614 . . . . . . 7  |-  ( ( k  e.  NN  /\  1  e.  NN )  ->  if ( k  e. 
Prime ,  k , 
1 )  e.  NN )
6219, 60, 61sylancl 643 . . . . . 6  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  NN )
6362nnrpd 10405 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  RR+ )
6463reeflogd 19991 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  ( log `  if ( k  e.  Prime ,  k ,  1 ) ) )  =  if ( k  e.  Prime ,  k ,  1 ) )
65 fvif 5556 . . . . . . 7  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  ( log `  1 ) )
66 log1 19955 . . . . . . . 8  |-  ( log `  1 )  =  0
67 ifeq2 3583 . . . . . . . 8  |-  ( ( log `  1 )  =  0  ->  if ( k  e.  Prime ,  ( log `  k
) ,  ( log `  1 ) )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
6866, 67ax-mp 8 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  ( log `  1
) )  =  if ( k  e.  Prime ,  ( log `  k
) ,  0 )
6965, 68eqtri 2316 . . . . . 6  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )
7047, 69syl6eqr 2346 . . . . 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 ) ) )
7170fveq2d 5545 . . . 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 ) ) ) )
72 id 19 . . . . . . 7  |-  ( n  =  k  ->  n  =  k )
73 eqidd 2297 . . . . . . 7  |-  ( n  =  k  ->  1  =  1 )
7437, 72, 73ifbieq12d 3600 . . . . . 6  |-  ( n  =  k  ->  if ( n  e.  Prime ,  n ,  1 )  =  if ( k  e.  Prime ,  k ,  1 ) )
75 prmorcht.1 . . . . . 6  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  n ,  1 ) )
76 vex 2804 . . . . . . 7  |-  k  e. 
_V
7760elexi 2810 . . . . . . 7  |-  1  e.  _V
7876, 77ifex 3636 . . . . . 6  |-  if ( k  e.  Prime ,  k ,  1 )  e. 
_V
7974, 75, 78fvmpt 5618 . . . . 5  |-  ( k  e.  NN  ->  ( F `  k )  =  if ( k  e. 
Prime ,  k , 
1 ) )
8019, 79syl 15 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( F `  k )  =  if ( k  e.  Prime ,  k ,  1 ) )
8164, 71, 803eqtr4d 2338 . . 3  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) `  k ) )  =  ( F `
 k ) )
8256, 57, 49, 59, 81seqhomo 11109 . 2  |-  ( A  e.  NN  ->  ( exp `  (  seq  1
(  +  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) )  =  (  seq  1 (  x.  ,  F ) `  A ) )
8354, 82eqtrd 2328 1  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  (  seq  1 (  x.  ,  F ) `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   ifcif 3578    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   NNcn 9762   2c2 9811   ZZcz 10040   ZZ>=cuz 10246   [,]cicc 10675   ...cfz 10798   |_cfl 10940    seq cseq 11062   sum_csu 12174   expce 12359   Primecprime 12774   logclog 19928   thetaccht 20344
This theorem is referenced by:  chtublem  20466  bposlem6  20544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-prm 12775  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cht 20350
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