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Theorem pntrval 26816
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 6825 . . 3 (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴))
2 id 22 . . 3 (𝑎 = 𝐴𝑎 = 𝐴)
31, 2oveq12d 7355 . 2 (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴))
4 pntrval.r . 2 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
5 ovex 7370 . 2 ((ψ‘𝐴) − 𝐴) ∈ V
63, 4, 5fvmpt 6931 1 (𝐴 ∈ ℝ+ → (𝑅𝐴) = ((ψ‘𝐴) − 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  cmpt 5175  cfv 6479  (class class class)co 7337  cmin 11306  +crp 12831  ψcchp 26348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-ov 7340
This theorem is referenced by:  pntrmax  26818  pntrsumo1  26819  selbergr  26822  selberg3r  26823  selberg4r  26824  pntrlog2bndlem2  26832  pntrlog2bndlem4  26834  pntrlog2bnd  26838  pntpbnd1a  26839  pntibndlem2  26845  pntlem3  26863
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