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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 6191 | . . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴)) | |
2 | 1 | oveq2d 6666 | . . 3 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴))) |
3 | 2 | sumeq1d 14431 | . 2 ⊢ (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛)) |
4 | df-chp 24825 | . 2 ⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛)) | |
5 | sumex 14418 | . 2 ⊢ Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛) ∈ V | |
6 | 3, 4, 5 | fvmpt 6282 | 1 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 ...cfz 12326 ⌊cfl 12591 Σcsu 14416 Λcvma 24818 ψcchp 24819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fun 5890 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 df-sum 14417 df-chp 24825 |
This theorem is referenced by: efchpcl 24851 chpfl 24876 chpp1 24881 chpwordi 24883 chp1 24893 chtlepsi 24931 chpval2 24943 vmadivsum 25171 selberg 25237 selberg3lem1 25246 selberg4 25250 pntsval2 25265 chpvalz 30706 |
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