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Mirrors > Home > MPE Home > Th. List > chpval | Structured version Visualization version GIF version |
Description: Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
Ref | Expression |
---|---|
chpval | ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴)) | |
2 | 1 | oveq2d 6565 | . . 3 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴))) |
3 | 2 | sumeq1d 14279 | . 2 ⊢ (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛)) |
4 | df-chp 24625 | . 2 ⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛)) | |
5 | sumex 14266 | . 2 ⊢ Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛) ∈ V | |
6 | 3, 4, 5 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 1c1 9816 ...cfz 12197 ⌊cfl 12453 Σcsu 14264 Λcvma 24618 ψcchp 24619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-sum 14265 df-chp 24625 |
This theorem is referenced by: efchpcl 24651 chpfl 24676 chpp1 24681 chpwordi 24683 chp1 24693 chtlepsi 24731 chpval2 24743 vmadivsum 24971 selberg 25037 selberg3lem1 25046 selberg4 25050 pntsval2 25065 |
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