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

Theorem chpval 27165
Description: Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
chpval (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (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 6906 . . . 4 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
21oveq2d 7447 . . 3 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
32sumeq1d 15736 . 2 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
4 df-chp 27142 . 2 ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
5 sumex 15724 . 2 Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛) ∈ V
63, 4, 5fvmpt 7016 1 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  cr 11154  1c1 11156  ...cfz 13547  cfl 13830  Σcsu 15722  Λcvma 27135  ψcchp 27136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seq 14043  df-sum 15723  df-chp 27142
This theorem is referenced by:  efchpcl  27168  chpfl  27193  chpp1  27198  chpwordi  27200  chp1  27210  chtlepsi  27250  chpval2  27262  vmadivsum  27526  selberg  27592  selberg3lem1  27601  selberg4  27605  pntsval2  27620  chpvalz  34643
  Copyright terms: Public domain W3C validator