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Theorem prmorcht 20412
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 ) )
Dummy variables  k  p are mutually distinct and distinct from all other variables.

Proof of Theorem prmorcht
StepHypRef Expression
1 nnre 9750 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  RR )
2 chtval 20344 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ k  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  k ) )
31, 2syl 17 . . . . . 6  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  k ) )
4 2nn 9874 . . . . . . . . . . 11  |-  2  e.  NN
5 nnuz 10260 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
64, 5eleqtri 2358 . . . . . . . . . 10  |-  2  e.  ( ZZ>= `  1 )
7 ppisval2 20338 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ( ZZ>= ` 
1 ) )  -> 
( ( 0 [,] A )  i^i  Prime )  =  ( ( 1 ... ( |_ `  A ) )  i^i 
Prime ) )
81, 6, 7sylancl 645 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... ( |_ `  A
) )  i^i  Prime ) )
9 nnz 10042 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  ZZ )
10 flid 10935 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
119, 10syl 17 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( |_ `  A )  =  A )
1211oveq2d 5837 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
1 ... ( |_ `  A ) )  =  ( 1 ... A
) )
1312ineq1d 3372 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 1 ... ( |_ `  A ) )  i^i  Prime )  =  ( ( 1 ... A
)  i^i  Prime ) )
148, 13eqtrd 2318 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... A )  i^i  Prime ) )
1514sumeq1d 12170 . . . . . . 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 3392 . . . . . . . 8  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
1716sseli 3179 . . . . . . . . . 10  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  ->  k  e.  ( 1 ... A
) )
18 elfznn 10815 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 1 ... A )  ->  k  e.  NN )
1918adantl 454 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  NN )
2019nnrpd 10386 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  RR+ )
2120relogcld 19970 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  RR )
2221recnd 8858 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  CC )
2317, 22sylan2 462 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  k  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  k )  e.  CC )
2423ralrimiva 2629 . . . . . . . 8  |-  ( A  e.  NN  ->  A. k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
)  e.  CC )
25 fzfi 11030 . . . . . . . . . 10  |-  ( 1 ... A )  e. 
Fin
2625olci 382 . . . . . . . . 9  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
27 sumss2 12195 . . . . . . . . 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 654 . . . . . . . 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 646 . . . . . . 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 2318 . . . . . 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 2318 . . . . 5  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  k
) ,  0 ) )
32 elin 3361 . . . . . . . 8  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( k  e.  ( 1 ... A
)  /\  k  e.  Prime ) )
3332baibr 874 . . . . . . 7  |-  ( k  e.  ( 1 ... A )  ->  (
k  e.  Prime  <->  k  e.  ( ( 1 ... A )  i^i  Prime ) ) )
3433ifbid 3586 . . . . . 6  |-  ( k  e.  ( 1 ... A )  ->  if ( k  e.  Prime ,  ( log `  k
) ,  0 )  =  if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 ) )
3534sumeq2i 12168 . . . . 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 2336 . . . 4  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e. 
Prime ,  ( log `  k ) ,  0 ) )
37 eleq1 2346 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
38 fveq2 5487 . . . . . . . 8  |-  ( n  =  k  ->  ( log `  n )  =  ( log `  k
) )
39 eqidd 2287 . . . . . . . 8  |-  ( n  =  k  ->  0  =  0 )
4037, 38, 39ifbieq12d 3590 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( log `  n
) ,  0 )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
41 eqid 2286 . . . . . . 7  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )
42 fvex 5501 . . . . . . . 8  |-  ( log `  k )  e.  _V
43 0cn 8828 . . . . . . . . 9  |-  0  e.  CC
4443elexi 2800 . . . . . . . 8  |-  0  e.  _V
4542, 44ifex 3626 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e. 
_V
4640, 41, 45fvmpt 5565 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) `  k )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
4719, 46syl 17 . . . . 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 10261 . . . . . 6  |-  ( A  e.  NN  <->  A  e.  ( ZZ>= `  1 )
)
4948biimpi 188 . . . . 5  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  1 )
)
50 ifcl 3604 . . . . . 6  |-  ( ( ( log `  k
)  e.  CC  /\  0  e.  CC )  ->  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )  e.  CC )
5122, 43, 50sylancl 645 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e.  CC )
5247, 49, 51fsumser 12199 . . . 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 2318 . . 3  |-  ( A  e.  NN  ->  ( theta `  A )  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) ) `  A ) )
5453fveq2d 5491 . 2  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  ( exp `  (  seq  1 (  +  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) ) )
55 addcl 8816 . . . 4  |-  ( ( k  e.  CC  /\  p  e.  CC )  ->  ( k  +  p
)  e.  CC )
5655adantl 454 . . 3  |-  ( ( A  e.  NN  /\  ( k  e.  CC  /\  p  e.  CC ) )  ->  ( k  +  p )  e.  CC )
5747, 51eqeltrd 2360 . . 3  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) `
 k )  e.  CC )
58 efadd 12371 . . . 4  |-  ( ( k  e.  CC  /\  p  e.  CC )  ->  ( exp `  (
k  +  p ) )  =  ( ( exp `  k )  x.  ( exp `  p
) ) )
5958adantl 454 . . 3  |-  ( ( A  e.  NN  /\  ( k  e.  CC  /\  p  e.  CC ) )  ->  ( exp `  ( k  +  p
) )  =  ( ( exp `  k
)  x.  ( exp `  p ) ) )
60 1nn 9754 . . . . . . 7  |-  1  e.  NN
61 ifcl 3604 . . . . . . 7  |-  ( ( k  e.  NN  /\  1  e.  NN )  ->  if ( k  e. 
Prime ,  k , 
1 )  e.  NN )
6219, 60, 61sylancl 645 . . . . . 6  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  NN )
6362nnrpd 10386 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  RR+ )
6463reeflogd 19971 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  ( log `  if ( k  e.  Prime ,  k ,  1 ) ) )  =  if ( k  e.  Prime ,  k ,  1 ) )
65 fvif 5502 . . . . . . 7  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  ( log `  1 ) )
66 log1 19935 . . . . . . . 8  |-  ( log `  1 )  =  0
67 ifeq2 3573 . . . . . . . 8  |-  ( ( log `  1 )  =  0  ->  if ( k  e.  Prime ,  ( log `  k
) ,  ( log `  1 ) )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
6866, 67ax-mp 10 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  ( log `  1
) )  =  if ( k  e.  Prime ,  ( log `  k
) ,  0 )
6965, 68eqtri 2306 . . . . . 6  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )
7047, 69syl6eqr 2336 . . . . 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 5491 . . . 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 21 . . . . . . 7  |-  ( n  =  k  ->  n  =  k )
73 eqidd 2287 . . . . . . 7  |-  ( n  =  k  ->  1  =  1 )
7437, 72, 73ifbieq12d 3590 . . . . . 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 2794 . . . . . . 7  |-  k  e. 
_V
7760elexi 2800 . . . . . . 7  |-  1  e.  _V
7876, 77ifex 3626 . . . . . 6  |-  if ( k  e.  Prime ,  k ,  1 )  e. 
_V
7974, 75, 78fvmpt 5565 . . . . 5  |-  ( k  e.  NN  ->  ( F `  k )  =  if ( k  e. 
Prime ,  k , 
1 ) )
8019, 79syl 17 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( F `  k )  =  if ( k  e.  Prime ,  k ,  1 ) )
8164, 71, 803eqtr4d 2328 . . 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 11089 . 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 2318 1  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  (  seq  1 (  x.  ,  F ) `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1625    e. wcel 1687   A.wral 2546    i^i cin 3154    C_ wss 3155   ifcif 3568    e. cmpt 4080   ` cfv 5223  (class class class)co 5821   Fincfn 6860   CCcc 8732   RRcr 8733   0cc0 8734   1c1 8735    + caddc 8737    x. cmul 8739   NNcn 9743   2c2 9792   ZZcz 10021   ZZ>=cuz 10227   [,]cicc 10655   ...cfz 10778   |_cfl 10920    seq cseq 11042   sum_csu 12154   expce 12339   Primecprime 12754   logclog 19908   thetaccht 20324
This theorem is referenced by:  chtublem  20446  bposlem6  20524
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812  ax-addf 8813  ax-mulf 8814
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-of 6041  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-er 6657  df-map 6771  df-pm 6772  df-ixp 6815  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-fi 7162  df-sup 7191  df-oi 7222  df-card 7569  df-cda 7791  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-4 9803  df-5 9804  df-6 9805  df-7 9806  df-8 9807  df-9 9808  df-10 9809  df-n0 9963  df-z 10022  df-dec 10122  df-uz 10228  df-q 10314  df-rp 10352  df-xneg 10449  df-xadd 10450  df-xmul 10451  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-sum 12155  df-ef 12345  df-sin 12347  df-cos 12348  df-pi 12350  df-dvds 12528  df-prm 12755  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-limc 19212  df-dv 19213  df-log 19910  df-cht 20330
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