Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6673 | . . 3 ⊢ (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴)) | |
2 | id 22 | . . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
3 | 1, 2 | oveq12d 7177 | . 2 ⊢ (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴)) |
4 | pntrval.r | . 2 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) | |
5 | ovex 7192 | . 2 ⊢ ((ψ‘𝐴) − 𝐴) ∈ V | |
6 | 3, 4, 5 | fvmpt 6771 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 − cmin 10873 ℝ+crp 12392 ψcchp 25673 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 |
This theorem is referenced by: pntrmax 26143 pntrsumo1 26144 selbergr 26147 selberg3r 26148 selberg4r 26149 pntrlog2bndlem2 26157 pntrlog2bndlem4 26159 pntrlog2bnd 26163 pntpbnd1a 26164 pntibndlem2 26170 pntlem3 26188 |
Copyright terms: Public domain | W3C validator |