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Theorem chpval 22458
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 5689 . . . 4  |-  ( x  =  A  ->  ( |_ `  x )  =  ( |_ `  A
) )
21oveq2d 6105 . . 3  |-  ( x  =  A  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... ( |_ `  A ) ) )
32sumeq1d 13176 . 2  |-  ( x  =  A  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) (Λ `  n )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) (Λ `  n )
)
4 df-chp 22434 . 2  |- ψ  =  ( x  e.  RR  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) (Λ `  n ) )
5 sumex 13163 . 2  |-  sum_ n  e.  ( 1 ... ( |_ `  A ) ) (Λ `  n )  e.  _V
63, 4, 5fvmpt 5772 1  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) (Λ `  n
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5416  (class class class)co 6089   RRcr 9279   1c1 9281   ...cfz 11435   |_cfl 11638   sum_csu 13161  Λcvma 22427  ψcchp 22428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-recs 6830  df-rdg 6864  df-seq 11805  df-sum 13162  df-chp 22434
This theorem is referenced by:  efchpcl  22461  chpfl  22486  chpp1  22491  chpwordi  22493  chp1  22503  chtlepsi  22543  chpval2  22555  vmadivsum  22729  selberg  22795  selberg3lem1  22804  selberg4  22808  pntsval2  22823
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