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Theorem chpval 25200
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 6411 . . . 4 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
21oveq2d 6894 . . 3 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
32sumeq1d 14772 . 2 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
4 df-chp 25177 . 2 ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
5 sumex 14759 . 2 Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛) ∈ V
63, 4, 5fvmpt 6507 1 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  cr 10223  1c1 10225  ...cfz 12580  cfl 12846  Σcsu 14757  Λcvma 25170  ψcchp 25171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-iota 6064  df-fun 6103  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-seq 13056  df-sum 14758  df-chp 25177
This theorem is referenced by:  efchpcl  25203  chpfl  25228  chpp1  25233  chpwordi  25235  chp1  25245  chtlepsi  25283  chpval2  25295  vmadivsum  25523  selberg  25589  selberg3lem1  25598  selberg4  25602  pntsval2  25617  chpvalz  31226
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