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Theorem chtval 23582
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 6278 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3685 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32sumeq1d 13605 . 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 23568 . 2  |-  theta  =  ( x  e.  RR  |->  sum_
p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p ) )
5 sumex 13592 . 2  |-  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  p
)  e.  _V
63, 4, 5fvmpt 5931 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 1398    e. wcel 1823    i^i cin 3460   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   [,]cicc 11535   sum_csu 13590   Primecprime 14301   logclog 23108   thetaccht 23562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-seq 12090  df-sum 13591  df-cht 23568
This theorem is referenced by:  efchtcl  23583  chtge0  23584  chtfl  23621  chtprm  23625  chtnprm  23626  chtwordi  23628  chtdif  23630  cht1  23637  prmorcht  23650  chtlepsi  23679  chtleppi  23683  chpchtsum  23692  chpub  23693  chtppilimlem1  23856
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