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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 6906 | . . 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 1540 ∈ wcel 2108 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 − cmin 11492 ℝ+crp 13034 ψcchp 27136 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: pntrmax 27608 pntrsumo1 27609 selbergr 27612 selberg3r 27613 selberg4r 27614 pntrlog2bndlem2 27622 pntrlog2bndlem4 27624 pntrlog2bnd 27628 pntpbnd1a 27629 pntibndlem2 27635 pntlem3 27653 |
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