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Theorem chpval 20360
Description: Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
chpval  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) (Λ `  n
) )
Distinct variable group:    A, n

Proof of Theorem chpval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( x  =  A  ->  ( |_ `  x )  =  ( |_ `  A
) )
21oveq2d 5874 . . 3  |-  ( x  =  A  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... ( |_ `  A ) ) )
32sumeq1d 12174 . 2  |-  ( x  =  A  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) (Λ `  n )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) (Λ `  n )
)
4 df-chp 20336 . 2  |- ψ  =  ( x  e.  RR  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) (Λ `  n ) )
5 sumex 12160 . 2  |-  sum_ n  e.  ( 1 ... ( |_ `  A ) ) (Λ `  n )  e.  _V
63, 4, 5fvmpt 5602 1  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) (Λ `  n
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738   ...cfz 10782   |_cfl 10924   sum_csu 12158  Λcvma 20329  ψcchp 20330
This theorem is referenced by:  efchpcl  20363  chpfl  20388  chpp1  20393  chpwordi  20395  chp1  20405  chtlepsi  20445  chpval2  20457  vmadivsum  20631  selberg  20697  selberg3lem1  20706  selberg4  20710  pntsval2  20725
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 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-sum 12159  df-chp 20336
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