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Theorem chpval 26423
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 6839 . . . 4 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
21oveq2d 7367 . . 3 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
32sumeq1d 15546 . 2 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
4 df-chp 26400 . 2 ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
5 sumex 15532 . 2 Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛) ∈ V
63, 4, 5fvmpt 6945 1 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351  cr 11008  1c1 11010  ...cfz 13378  cfl 13649  Σcsu 15530  Λcvma 26393  ψcchp 26394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-iota 6445  df-fun 6495  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-seq 13861  df-sum 15531  df-chp 26400
This theorem is referenced by:  efchpcl  26426  chpfl  26451  chpp1  26456  chpwordi  26458  chp1  26468  chtlepsi  26506  chpval2  26518  vmadivsum  26782  selberg  26848  selberg3lem1  26857  selberg4  26861  pntsval2  26876  chpvalz  33053
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