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Theorem chtval 22582
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 6209 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3660 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32sumeq1d 13297 . 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 22568 . 2  |-  theta  =  ( x  e.  RR  |->  sum_
p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p ) )
5 sumex 13284 . 2  |-  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  p
)  e.  _V
63, 4, 5fvmpt 5884 1  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    i^i cin 3436   ` cfv 5527  (class class class)co 6201   RRcr 9393   0cc0 9394   [,]cicc 11415   sum_csu 13282   Primecprime 13882   logclog 22140   thetaccht 22562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-recs 6943  df-rdg 6977  df-seq 11925  df-sum 13283  df-cht 22568
This theorem is referenced by:  efchtcl  22583  chtge0  22584  chtfl  22621  chtprm  22625  chtnprm  22626  chtwordi  22628  chtdif  22630  cht1  22637  prmorcht  22650  chtlepsi  22679  chtleppi  22683  chpchtsum  22692  chpub  22693  chtppilimlem1  22856
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