Theorem pntrval 27480

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 6861	. . 3 ⊢ (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴))
2		id 22	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
3	1, 2	oveq12d 7408	. 2 ⊢ (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴))
4		pntrval.r	. 2 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
5		ovex 7423	. 2 ⊢ ((ψ‘𝐴) − 𝐴) ∈ V
6	3, 4, 5	fvmpt 6971	1 ⊢ (𝐴 ∈ ℝ+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390   − cmin 11412  ℝ⁺crp 12958  ψcchp 27010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393

This theorem is referenced by:  pntrmax  27482  pntrsumo1  27483  selbergr  27486  selberg3r  27487  selberg4r  27488  pntrlog2bndlem2  27496  pntrlog2bndlem4  27498  pntrlog2bnd  27502  pntpbnd1a  27503  pntibndlem2  27509  pntlem3  27527