Theorem pntrval 25051

Description: Define the residual of the second Chebyshev function. The goal is to have 𝑅(𝑥) ∈ 𝑜(𝑥), or 𝑅(𝑥) / 𝑥 ⇝_𝑟 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+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴))

Distinct variable group:   𝐴,𝑎

Allowed substitution hint:   𝑅(𝑎)

Proof of Theorem pntrval

Step	Hyp	Ref	Expression
1		fveq2 6103	. . 3 ⊢ (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴))
2		id 22	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
3	1, 2	oveq12d 6567	. 2 ⊢ (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴))
4		pntrval.r	. 2 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
5		ovex 6577	. 2 ⊢ ((ψ‘𝐴) − 𝐴) ∈ V
6	3, 4, 5	fvmpt 6191	1 ⊢ (𝐴 ∈ ℝ+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   − cmin 10145  ℝ⁺crp 11708  ψcchp 24619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552

This theorem is referenced by:  pntrmax  25053  pntrsumo1  25054  selbergr  25057  selberg3r  25058  selberg4r  25059  pntrlog2bndlem2  25067  pntrlog2bndlem4  25069  pntrlog2bnd  25073  pntpbnd1a  25074  pntibndlem2  25080  pntlem3  25098