Theorem pntrval 26140

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 6672	. . 3 ⊢ (𝑎 = 𝐴 → (ψ‘𝑎) = (ψ‘𝐴))
2		id 22	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
3	1, 2	oveq12d 7176	. 2 ⊢ (𝑎 = 𝐴 → ((ψ‘𝑎) − 𝑎) = ((ψ‘𝐴) − 𝐴))
4		pntrval.r	. 2 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
5		ovex 7191	. 2 ⊢ ((ψ‘𝐴) − 𝐴) ∈ V
6	3, 4, 5	fvmpt 6770	1 ⊢ (𝐴 ∈ ℝ+ → (𝑅‘𝐴) = ((ψ‘𝐴) − 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158   − cmin 10872  ℝ⁺crp 12392  ψcchp 25672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161

This theorem is referenced by:  pntrmax  26142  pntrsumo1  26143  selbergr  26146  selberg3r  26147  selberg4r  26148  pntrlog2bndlem2  26156  pntrlog2bndlem4  26158  pntrlog2bnd  26162  pntpbnd1a  26163  pntibndlem2  26169  pntlem3  26187