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Theorem chpval 26271
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 6774 . . . 4 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
21oveq2d 7291 . . 3 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
32sumeq1d 15413 . 2 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
4 df-chp 26248 . 2 ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
5 sumex 15399 . 2 Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛) ∈ V
63, 4, 5fvmpt 6875 1 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cfv 6433  (class class class)co 7275  cr 10870  1c1 10872  ...cfz 13239  cfl 13510  Σcsu 15397  Λcvma 26241  ψcchp 26242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-iota 6391  df-fun 6435  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seq 13722  df-sum 15398  df-chp 26248
This theorem is referenced by:  efchpcl  26274  chpfl  26299  chpp1  26304  chpwordi  26306  chp1  26316  chtlepsi  26354  chpval2  26366  vmadivsum  26630  selberg  26696  selberg3lem1  26705  selberg4  26709  pntsval2  26724  chpvalz  32608
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