Theorem chpval 25851

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.

Step	Hyp	Ref	Expression
1		fveq2 6668	. . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
2	1	oveq2d 7180	. . 3 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
3	2	sumeq1d 15144	. 2 ⊢ (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))
4		df-chp 25828	. 2 ⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
5		sumex 15130	. 2 ⊢ Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛) ∈ V
6	3, 4, 5	fvmpt 6769	1 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2113  ‘cfv 6333  (class class class)co 7164  ℝcr 10607  1c1 10609  ...cfz 12974  ⌊cfl 13244  Σcsu 15128  Λcvma 25821  ψcchp 25822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-iota 6291  df-fun 6335  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-seq 13454  df-sum 15129  df-chp 25828

This theorem is referenced by:  efchpcl  25854  chpfl  25879  chpp1  25884  chpwordi  25886  chp1  25896  chtlepsi  25934  chpval2  25946  vmadivsum  26210  selberg  26276  selberg3lem1  26285  selberg4  26289  pntsval2  26304  chpvalz  32170