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Theorem pntrval 27445
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 6884 . . 3 (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴))
2 id 22 . . 3 (𝑎 = 𝐴𝑎 = 𝐴)
31, 2oveq12d 7422 . 2 (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴))
4 pntrval.r . 2 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
5 ovex 7437 . 2 ((ψ‘𝐴) − 𝐴) ∈ V
63, 4, 5fvmpt 6991 1 (𝐴 ∈ ℝ+ → (𝑅𝐴) = ((ψ‘𝐴) − 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cmpt 5224  cfv 6536  (class class class)co 7404  cmin 11445  +crp 12977  ψcchp 26975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407
This theorem is referenced by:  pntrmax  27447  pntrsumo1  27448  selbergr  27451  selberg3r  27452  selberg4r  27453  pntrlog2bndlem2  27461  pntrlog2bndlem4  27463  pntrlog2bnd  27467  pntpbnd1a  27468  pntibndlem2  27474  pntlem3  27492
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