Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 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)) |
Colors of variables: wff setvar class |
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 |
Copyright terms: Public domain | W3C validator |