MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chpval Unicode version

Theorem chpval 20376
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 5541 . . . 4  |-  ( x  =  A  ->  ( |_ `  x )  =  ( |_ `  A
) )
21oveq2d 5890 . . 3  |-  ( x  =  A  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... ( |_ `  A ) ) )
32sumeq1d 12190 . 2  |-  ( x  =  A  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) (Λ `  n )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) (Λ `  n )
)
4 df-chp 20352 . 2  |- ψ  =  ( x  e.  RR  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) (Λ `  n ) )
5 sumex 12176 . 2  |-  sum_ n  e.  ( 1 ... ( |_ `  A ) ) (Λ `  n )  e.  _V
63, 4, 5fvmpt 5618 1  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) (Λ `  n
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754   ...cfz 10798   |_cfl 10940   sum_csu 12174  Λcvma 20345  ψcchp 20346
This theorem is referenced by:  efchpcl  20379  chpfl  20404  chpp1  20409  chpwordi  20411  chp1  20421  chtlepsi  20461  chpval2  20473  vmadivsum  20647  selberg  20713  selberg3lem1  20722  selberg4  20726  pntsval2  20741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-sum 12175  df-chp 20352
  Copyright terms: Public domain W3C validator