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Theorem chtval 20344
Description: Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
chtval  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
Distinct variable group:    A, p

Proof of Theorem chtval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 5828 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3370 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32sumeq1d 12170 . 2  |-  ( x  =  A  ->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)  =  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  p
) )
4 df-cht 20330 . 2  |-  theta  =  ( x  e.  RR  |->  sum_
p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p ) )
5 sumex 12156 . 2  |-  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  p
)  e.  _V
63, 4, 5fvmpt 5564 1  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1685    i^i cin 3152   ` cfv 5221  (class class class)co 5820   RRcr 8732   0cc0 8733   [,]cicc 10655   sum_csu 12154   Primecprime 12754   logclog 19908   thetaccht 20324
This theorem is referenced by:  efchtcl  20345  chtge0  20346  chtfl  20383  chtprm  20387  chtnprm  20388  chtwordi  20390  chtdif  20392  cht1  20399  prmorcht  20412  chtlepsi  20441  chtleppi  20445  chpchtsum  20454  chpub  20455  chtppilimlem1  20618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-recs 6384  df-rdg 6419  df-seq 11043  df-sum 12155  df-cht 20330
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