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Mirrors > Home > MPE Home > Th. List > pntrval | Structured version Visualization version GIF version |
Description: Define the residual of the second Chebyshev function. The goal is to have 𝑅(𝑥) ∈ 𝑜(𝑥), or 𝑅(𝑥) / 𝑥 ⇝𝑟 0. (Contributed by Mario Carneiro, 8-Apr-2016.) |
Ref | Expression |
---|---|
pntrval.r | ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) |
Ref | Expression |
---|---|
pntrval | ⊢ (𝐴 ∈ ℝ+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6907 | . . 3 ⊢ (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴)) | |
2 | id 22 | . . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
3 | 1, 2 | oveq12d 7449 | . 2 ⊢ (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴)) |
4 | pntrval.r | . 2 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) | |
5 | ovex 7464 | . 2 ⊢ ((ψ‘𝐴) − 𝐴) ∈ V | |
6 | 3, 4, 5 | fvmpt 7016 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 − cmin 11490 ℝ+crp 13032 ψcchp 27151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 |
This theorem is referenced by: pntrmax 27623 pntrsumo1 27624 selbergr 27627 selberg3r 27628 selberg4r 27629 pntrlog2bndlem2 27637 pntrlog2bndlem4 27639 pntrlog2bnd 27643 pntpbnd1a 27644 pntibndlem2 27650 pntlem3 27668 |
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