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Theorem prmorcht 20378
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
StepHypRef Expression
1 nnre 9721 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  RR )
2 chtval 20310 . . . . . . 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 9844 . . . . . . . . . . 11  |-  2  e.  NN
5 nnuz 10230 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
64, 5eleqtri 2330 . . . . . . . . . 10  |-  2  e.  ( ZZ>= `  1 )
7 ppisval2 20304 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ( ZZ>= ` 
1 ) )  -> 
( ( 0 [,] A )  i^i  Prime )  =  ( ( 1 ... ( |_ `  A ) )  i^i 
Prime ) )
81, 6, 7sylancl 646 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... ( |_ `  A
) )  i^i  Prime ) )
9 nnz 10012 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  ZZ )
10 flid 10905 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
119, 10syl 17 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( |_ `  A )  =  A )
1211oveq2d 5808 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
1 ... ( |_ `  A ) )  =  ( 1 ... A
) )
1312ineq1d 3344 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 1 ... ( |_ `  A ) )  i^i  Prime )  =  ( ( 1 ... A
)  i^i  Prime ) )
148, 13eqtrd 2290 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 1 ... A )  i^i  Prime ) )
1514sumeq1d 12139 . . . . . . 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 3364 . . . . . . . 8  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
1716sseli 3151 . . . . . . . . . 10  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  ->  k  e.  ( 1 ... A
) )
18 elfznn 10785 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 1 ... A )  ->  k  e.  NN )
1918adantl 454 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  NN )
2019nnrpd 10356 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  k  e.  RR+ )
2120relogcld 19936 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( log `  k
)  e.  RR )
2221recnd 8829 . . . . . . . . . 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 2601 . . . . . . . 8  |-  ( A  e.  NN  ->  A. k  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  k
)  e.  CC )
25 fzfi 11000 . . . . . . . . . 10  |-  ( 1 ... A )  e. 
Fin
2625olci 382 . . . . . . . . 9  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
27 sumss2 12164 . . . . . . . . 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 655 . . . . . . . 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 647 . . . . . . 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 2290 . . . . . 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 2290 . . . . 5  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  k
) ,  0 ) )
32 elin 3333 . . . . . . . 8  |-  ( k  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( k  e.  ( 1 ... A
)  /\  k  e.  Prime ) )
3332baibr 877 . . . . . . 7  |-  ( k  e.  ( 1 ... A )  ->  (
k  e.  Prime  <->  k  e.  ( ( 1 ... A )  i^i  Prime ) ) )
3433ifbid 3557 . . . . . 6  |-  ( k  e.  ( 1 ... A )  ->  if ( k  e.  Prime ,  ( log `  k
) ,  0 )  =  if ( k  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  k ) ,  0 ) )
3534sumeq2i 12137 . . . . 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 2308 . . . 4  |-  ( A  e.  NN  ->  ( theta `  A )  = 
sum_ k  e.  ( 1 ... A ) if ( k  e. 
Prime ,  ( log `  k ) ,  0 ) )
37 eleq1 2318 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  Prime  <->  k  e.  Prime ) )
38 fveq2 5458 . . . . . . . 8  |-  ( n  =  k  ->  ( log `  n )  =  ( log `  k
) )
39 eqidd 2259 . . . . . . . 8  |-  ( n  =  k  ->  0  =  0 )
4037, 38, 39ifbieq12d 3561 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  Prime ,  ( log `  n
) ,  0 )  =  if ( k  e.  Prime ,  ( log `  k ) ,  0 ) )
41 eqid 2258 . . . . . . 7  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) )
42 fvex 5472 . . . . . . . 8  |-  ( log `  k )  e.  _V
43 0cn 8799 . . . . . . . . 9  |-  0  e.  CC
4443elexi 2772 . . . . . . . 8  |-  0  e.  _V
4542, 44ifex 3597 . . . . . . 7  |-  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e. 
_V
4640, 41, 45fvmpt 5536 . . . . . 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 10231 . . . . . 6  |-  ( A  e.  NN  <->  A  e.  ( ZZ>= `  1 )
)
4948biimpi 188 . . . . 5  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  1 )
)
50 ifcl 3575 . . . . . 6  |-  ( ( ( log `  k
)  e.  CC  /\  0  e.  CC )  ->  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )  e.  CC )
5122, 43, 50sylancl 646 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  ( log `  k ) ,  0 )  e.  CC )
5247, 49, 51fsumser 12168 . . . 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 2290 . . 3  |-  ( A  e.  NN  ->  ( theta `  A )  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) ) `  A ) )
5453fveq2d 5462 . 2  |-  ( A  e.  NN  ->  ( exp `  ( theta `  A
) )  =  ( exp `  (  seq  1 (  +  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n
) ,  0 ) ) ) `  A
) ) )
55 addcl 8787 . . . 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 2332 . . 3  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( log `  n ) ,  0 ) ) `
 k )  e.  CC )
58 efadd 12337 . . . 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 9725 . . . . . . 7  |-  1  e.  NN
61 ifcl 3575 . . . . . . 7  |-  ( ( k  e.  NN  /\  1  e.  NN )  ->  if ( k  e. 
Prime ,  k , 
1 )  e.  NN )
6219, 60, 61sylancl 646 . . . . . 6  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  NN )
6362nnrpd 10356 . . . . 5  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  if ( k  e.  Prime ,  k ,  1 )  e.  RR+ )
6463reeflogd 19937 . . . 4  |-  ( ( A  e.  NN  /\  k  e.  ( 1 ... A ) )  ->  ( exp `  ( log `  if ( k  e.  Prime ,  k ,  1 ) ) )  =  if ( k  e.  Prime ,  k ,  1 ) )
65 fvif 5473 . . . . . . 7  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  ( log `  1 ) )
66 log1 19901 . . . . . . . 8  |-  ( log `  1 )  =  0
67 ifeq2 3544 . . . . . . . 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 2278 . . . . . 6  |-  ( log `  if ( k  e. 
Prime ,  k , 
1 ) )  =  if ( k  e. 
Prime ,  ( log `  k ) ,  0 )
7047, 69syl6eqr 2308 . . . . 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 5462 . . . 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 2259 . . . . . . 7  |-  ( n  =  k  ->  1  =  1 )
7437, 72, 73ifbieq12d 3561 . . . . . 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 2766 . . . . . . 7  |-  k  e. 
_V
7760elexi 2772 . . . . . . 7  |-  1  e.  _V
7876, 77ifex 3597 . . . . . 6  |-  if ( k  e.  Prime ,  k ,  1 )  e. 
_V
7974, 75, 78fvmpt 5536 . . . . 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 2300 . . 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 11059 . 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 2290 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 1619    e. wcel 1621   A.wral 2518    i^i cin 3126    C_ wss 3127   ifcif 3539    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   Fincfn 6831   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710   NNcn 9714   2c2 9763   ZZcz 9991   ZZ>=cuz 10197   [,]cicc 10625   ...cfz 10748   |_cfl 10890    seq cseq 11012   sum_csu 12123   expce 12305   Primecprime 12720   logclog 19874   thetaccht 20290
This theorem is referenced by:  chtublem  20412  bposlem6  20490
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-q 10284  df-rp 10322  df-xneg 10419  df-xadd 10420  df-xmul 10421  df-ioo 10626  df-ioc 10627  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-fl 10891  df-mod 10940  df-seq 11013  df-exp 11071  df-fac 11255  df-bc 11282  df-hash 11304  df-shft 11527  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-limsup 11910  df-clim 11927  df-rlim 11928  df-sum 12124  df-ef 12311  df-sin 12313  df-cos 12314  df-pi 12316  df-divides 12494  df-prime 12721  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-starv 13185  df-sca 13186  df-vsca 13187  df-tset 13189  df-ple 13190  df-ds 13192  df-hom 13194  df-cco 13195  df-rest 13289  df-topn 13290  df-topgen 13306  df-pt 13307  df-prds 13310  df-xrs 13365  df-0g 13366  df-gsum 13367  df-qtop 13372  df-imas 13373  df-xps 13375  df-mre 13450  df-mrc 13451  df-acs 13453  df-mnd 14329  df-submnd 14378  df-mulg 14454  df-cntz 14755  df-cmn 15053  df-xmet 16335  df-met 16336  df-bl 16337  df-mopn 16338  df-cnfld 16340  df-top 16598  df-bases 16600  df-topon 16601  df-topsp 16602  df-cld 16718  df-ntr 16719  df-cls 16720  df-nei 16797  df-lp 16830  df-perf 16831  df-cn 16919  df-cnp 16920  df-haus 17005  df-tx 17219  df-hmeo 17408  df-fbas 17482  df-fg 17483  df-fil 17503  df-fm 17595  df-flim 17596  df-flf 17597  df-xms 17847  df-ms 17848  df-tms 17849  df-cncf 18344  df-limc 19178  df-dv 19179  df-log 19876  df-cht 20296
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