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Theorem chtval 20854
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 6056 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3509 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32sumeq1d 12458 . 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 20840 . 2  |-  theta  =  ( x  e.  RR  |->  sum_
p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p ) )
5 sumex 12444 . 2  |-  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  p
)  e.  _V
63, 4, 5fvmpt 5773 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 1649    e. wcel 1721    i^i cin 3287   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954   [,]cicc 10883   sum_csu 12442   Primecprime 13042   logclog 20413   thetaccht 20834
This theorem is referenced by:  efchtcl  20855  chtge0  20856  chtfl  20893  chtprm  20897  chtnprm  20898  chtwordi  20900  chtdif  20902  cht1  20909  prmorcht  20922  chtlepsi  20951  chtleppi  20955  chpchtsum  20964  chpub  20965  chtppilimlem1  21128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-seq 11287  df-sum 12443  df-cht 20840
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