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Theorem pntrval 27621
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 6907 . . 3 (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴))
2 id 22 . . 3 (𝑎 = 𝐴𝑎 = 𝐴)
31, 2oveq12d 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
63, 4, 5fvmpt 7016 1 (𝐴 ∈ ℝ+ → (𝑅𝐴) = ((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  +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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