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Theorem pntrval 25151
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 6148 . . 3 (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴))
2 id 22 . . 3 (𝑎 = 𝐴𝑎 = 𝐴)
31, 2oveq12d 6622 . 2 (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴))
4 pntrval.r . 2 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
5 ovex 6632 . 2 ((ψ‘𝐴) − 𝐴) ∈ V
63, 4, 5fvmpt 6239 1 (𝐴 ∈ ℝ+ → (𝑅𝐴) = ((ψ‘𝐴) − 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cmpt 4673  cfv 5847  (class class class)co 6604  cmin 10210  +crp 11776  ψcchp 24719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607
This theorem is referenced by:  pntrmax  25153  pntrsumo1  25154  selbergr  25157  selberg3r  25158  selberg4r  25159  pntrlog2bndlem2  25167  pntrlog2bndlem4  25169  pntrlog2bnd  25173  pntpbnd1a  25174  pntibndlem2  25180  pntlem3  25198
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