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Theorem pntrval 25671
Description: Define the residual of the second Chebyshev function. The goal is to have 𝑅(𝑥) ∈ 𝑜(𝑥), or 𝑅(𝑥) / 𝑥𝑟 0. (Contributed by Mario Carneiro, 8-Apr-2016.)
Hypothesis
Ref Expression
pntrval.r 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
Assertion
Ref Expression
pntrval (𝐴 ∈ ℝ+ → (𝑅𝐴) = ((ψ‘𝐴) − 𝐴))
Distinct variable group:   𝐴,𝑎
Allowed substitution hint:   𝑅(𝑎)

Proof of Theorem pntrval
StepHypRef Expression
1 fveq2 6437 . . 3 (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴))
2 id 22 . . 3 (𝑎 = 𝐴𝑎 = 𝐴)
31, 2oveq12d 6928 . 2 (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴))
4 pntrval.r . 2 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
5 ovex 6942 . 2 ((ψ‘𝐴) − 𝐴) ∈ V
63, 4, 5fvmpt 6533 1 (𝐴 ∈ ℝ+ → (𝑅𝐴) = ((ψ‘𝐴) − 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  cmpt 4954  cfv 6127  (class class class)co 6910  cmin 10592  +crp 12119  ψcchp 25239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913
This theorem is referenced by:  pntrmax  25673  pntrsumo1  25674  selbergr  25677  selberg3r  25678  selberg4r  25679  pntrlog2bndlem2  25687  pntrlog2bndlem4  25689  pntrlog2bnd  25693  pntpbnd1a  25694  pntibndlem2  25700  pntlem3  25718
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