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Theorem selberg3r 27406
Description: Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.8 of [Shapiro], p. 429. (Contributed by Mario Carneiro, 30-May-2016.)
Hypothesis
Ref Expression
pntrval.r 𝑅 = (π‘Ž ∈ ℝ+ ↦ ((Οˆβ€˜π‘Ž) βˆ’ π‘Ž))
Assertion
Ref Expression
selberg3r (π‘₯ ∈ (1(,)+∞) ↦ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯)) ∈ 𝑂(1)
Distinct variable groups:   𝑛,π‘Ž,π‘₯   𝑅,𝑛,π‘₯
Allowed substitution hint:   𝑅(π‘Ž)

Proof of Theorem selberg3r
StepHypRef Expression
1 elioore 13350 . . . . . . . . . . . . 13 (π‘₯ ∈ (1(,)+∞) β†’ π‘₯ ∈ ℝ)
21adantl 481 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ ℝ)
3 1rp 12974 . . . . . . . . . . . . 13 1 ∈ ℝ+
43a1i 11 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ∈ ℝ+)
5 1red 11211 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ∈ ℝ)
6 eliooord 13379 . . . . . . . . . . . . . . 15 (π‘₯ ∈ (1(,)+∞) β†’ (1 < π‘₯ ∧ π‘₯ < +∞))
76adantl 481 . . . . . . . . . . . . . 14 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (1 < π‘₯ ∧ π‘₯ < +∞))
87simpld 494 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 < π‘₯)
95, 2, 8ltled 11358 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ≀ π‘₯)
102, 4, 9rpgecld 13051 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ ℝ+)
1110relogcld 26461 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ ℝ)
1211recnd 11238 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ β„‚)
13122timesd 12451 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (2 Β· (logβ€˜π‘₯)) = ((logβ€˜π‘₯) + (logβ€˜π‘₯)))
1413oveq2d 7417 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))) = (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ ((logβ€˜π‘₯) + (logβ€˜π‘₯))))
15 chpcl 26960 . . . . . . . . . . . . 13 (π‘₯ ∈ ℝ β†’ (Οˆβ€˜π‘₯) ∈ ℝ)
162, 15syl 17 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Οˆβ€˜π‘₯) ∈ ℝ)
1716, 11remulcld 11240 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) ∈ ℝ)
18 2re 12282 . . . . . . . . . . . . . 14 2 ∈ ℝ
1918a1i 11 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 2 ∈ ℝ)
202, 8rplogcld 26467 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ ℝ+)
2119, 20rerpdivcld 13043 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (2 / (logβ€˜π‘₯)) ∈ ℝ)
22 fzfid 13934 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (1...(βŒŠβ€˜π‘₯)) ∈ Fin)
23 elfznn 13526 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ β„•)
2423adantl 481 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ β„•)
25 vmacl 26954 . . . . . . . . . . . . . . . 16 (𝑛 ∈ β„• β†’ (Ξ›β€˜π‘›) ∈ ℝ)
2624, 25syl 17 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘›) ∈ ℝ)
272adantr 480 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ π‘₯ ∈ ℝ)
2827, 24nndivred 12262 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ ℝ)
29 chpcl 26960 . . . . . . . . . . . . . . . 16 ((π‘₯ / 𝑛) ∈ ℝ β†’ (Οˆβ€˜(π‘₯ / 𝑛)) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Οˆβ€˜(π‘₯ / 𝑛)) ∈ ℝ)
3126, 30remulcld 11240 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) ∈ ℝ)
3224nnrpd 13010 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ ℝ+)
3332relogcld 26461 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (logβ€˜π‘›) ∈ ℝ)
3431, 33remulcld 11240 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) ∈ ℝ)
3522, 34fsumrecl 15676 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) ∈ ℝ)
3621, 35remulcld 11240 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))) ∈ ℝ)
3717, 36readdcld 11239 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) ∈ ℝ)
3837, 10rerpdivcld 13043 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) ∈ ℝ)
3938recnd 11238 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) ∈ β„‚)
4039, 12, 12subsub4d 11598 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)) βˆ’ (logβ€˜π‘₯)) = (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ ((logβ€˜π‘₯) + (logβ€˜π‘₯))))
4114, 40eqtr4d 2767 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))) = ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)) βˆ’ (logβ€˜π‘₯)))
4241oveq1d 7416 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))) βˆ’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯))) = (((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)) βˆ’ (logβ€˜π‘₯)) βˆ’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯))))
4339, 12subcld 11567 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)) ∈ β„‚)
44 2cn 12283 . . . . . . . . 9 2 ∈ β„‚
4544a1i 11 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 2 ∈ β„‚)
4620rpne0d 13017 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) β‰  0)
4745, 12, 46divcld 11986 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (2 / (logβ€˜π‘₯)) ∈ β„‚)
4826, 24nndivred 12262 . . . . . . . . . 10 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) / 𝑛) ∈ ℝ)
4948, 33remulcld 11240 . . . . . . . . 9 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) ∈ ℝ)
5022, 49fsumrecl 15676 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) ∈ ℝ)
5150recnd 11238 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) ∈ β„‚)
5247, 51mulcld 11230 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) ∈ β„‚)
5343, 52, 12nnncan2d 11602 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)) βˆ’ (logβ€˜π‘₯)) βˆ’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯))) = ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)) βˆ’ ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))))
54 pntrval.r . . . . . . . . . . . . 13 𝑅 = (π‘Ž ∈ ℝ+ ↦ ((Οˆβ€˜π‘Ž) βˆ’ π‘Ž))
5554pntrf 27400 . . . . . . . . . . . 12 𝑅:ℝ+βŸΆβ„
5655ffvelcdmi 7075 . . . . . . . . . . 11 (π‘₯ ∈ ℝ+ β†’ (π‘…β€˜π‘₯) ∈ ℝ)
5710, 56syl 17 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘…β€˜π‘₯) ∈ ℝ)
5857recnd 11238 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘…β€˜π‘₯) ∈ β„‚)
5958, 12mulcld 11230 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) ∈ β„‚)
6036recnd 11238 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))) ∈ β„‚)
6159, 60addcld 11229 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) ∈ β„‚)
622recnd 11238 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ β„‚)
6362, 52mulcld 11230 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) ∈ β„‚)
6410rpne0d 13017 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ β‰  0)
6561, 63, 62, 64divsubdird 12025 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))) / π‘₯) = (((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ ((π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) / π‘₯)))
6659, 60, 63addsubassd 11587 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))) = (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))) βˆ’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))))))
6735recnd 11238 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) ∈ β„‚)
6862, 51mulcld 11230 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) ∈ β„‚)
6947, 67, 68subdid 11666 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 / (logβ€˜π‘₯)) Β· (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))) = (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))) βˆ’ ((2 / (logβ€˜π‘₯)) Β· (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))))
7049recnd 11238 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) ∈ β„‚)
7122, 62, 70fsummulc2 15726 . . . . . . . . . . . . . 14 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))
7271oveq2d 7417 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))))
7334recnd 11238 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) ∈ β„‚)
7462adantr 480 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ π‘₯ ∈ β„‚)
7574, 70mulcld 11230 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) ∈ β„‚)
7622, 73, 75fsumsub 15730 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))))
7772, 76eqtr4d 2767 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))))
7826recnd 11238 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘›) ∈ β„‚)
7930recnd 11238 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Οˆβ€˜(π‘₯ / 𝑛)) ∈ β„‚)
8033recnd 11238 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (logβ€˜π‘›) ∈ β„‚)
8178, 79, 80mul32d 11420 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) = (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (Οˆβ€˜(π‘₯ / 𝑛))))
8224nncnd 12224 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ β„‚)
8324nnne0d 12258 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 β‰  0)
8478, 80, 82, 83div23d 12023 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) / 𝑛) = (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))
8584oveq2d 7417 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ Β· (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) / 𝑛)) = (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))
8678, 80mulcld 11230 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) ∈ β„‚)
8774, 86, 82, 83div12d 12022 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ Β· (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) / 𝑛)) = (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘₯ / 𝑛)))
8885, 87eqtr3d 2766 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) = (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘₯ / 𝑛)))
8981, 88oveq12d 7419 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = ((((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (Οˆβ€˜(π‘₯ / 𝑛))) βˆ’ (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘₯ / 𝑛))))
9010adantr 480 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ π‘₯ ∈ ℝ+)
9190, 32rpdivcld 13029 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ ℝ+)
9254pntrval 27399 . . . . . . . . . . . . . . . . . 18 ((π‘₯ / 𝑛) ∈ ℝ+ β†’ (π‘…β€˜(π‘₯ / 𝑛)) = ((Οˆβ€˜(π‘₯ / 𝑛)) βˆ’ (π‘₯ / 𝑛)))
9391, 92syl 17 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘…β€˜(π‘₯ / 𝑛)) = ((Οˆβ€˜(π‘₯ / 𝑛)) βˆ’ (π‘₯ / 𝑛)))
9493oveq2d 7417 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘…β€˜(π‘₯ / 𝑛))) = (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· ((Οˆβ€˜(π‘₯ / 𝑛)) βˆ’ (π‘₯ / 𝑛))))
9528recnd 11238 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ β„‚)
9686, 79, 95subdid 11666 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· ((Οˆβ€˜(π‘₯ / 𝑛)) βˆ’ (π‘₯ / 𝑛))) = ((((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (Οˆβ€˜(π‘₯ / 𝑛))) βˆ’ (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘₯ / 𝑛))))
9794, 96eqtrd 2764 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘…β€˜(π‘₯ / 𝑛))) = ((((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (Οˆβ€˜(π‘₯ / 𝑛))) βˆ’ (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘₯ / 𝑛))))
9889, 97eqtr4d 2767 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘…β€˜(π‘₯ / 𝑛))))
9955ffvelcdmi 7075 . . . . . . . . . . . . . . . . 17 ((π‘₯ / 𝑛) ∈ ℝ+ β†’ (π‘…β€˜(π‘₯ / 𝑛)) ∈ ℝ)
10091, 99syl 17 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘…β€˜(π‘₯ / 𝑛)) ∈ ℝ)
101100recnd 11238 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘…β€˜(π‘₯ / 𝑛)) ∈ β„‚)
10278, 101, 80mul32d 11420 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) = (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) Β· (π‘…β€˜(π‘₯ / 𝑛))))
10398, 102eqtr4d 2767 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = (((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))
104103sumeq2dv 15645 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· (((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))
10577, 104eqtrd 2764 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))
106105oveq2d 7417 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 / (logβ€˜π‘₯)) Β· (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)) βˆ’ (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))) = ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))))
10747, 62, 51mul12d 11419 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 / (logβ€˜π‘₯)) Β· (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))))
108107oveq2d 7417 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))) βˆ’ ((2 / (logβ€˜π‘₯)) Β· (π‘₯ Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))) = (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))) βˆ’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))))
10969, 106, 1083eqtr3rd 2773 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))) βˆ’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))) = ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))))
110109oveq2d 7417 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›))) βˆ’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))))) = (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))))
11166, 110eqtrd 2764 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))) = (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))))
112111oveq1d 7416 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))) / π‘₯) = ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯))
11354pntrval 27399 . . . . . . . . . . . . . 14 (π‘₯ ∈ ℝ+ β†’ (π‘…β€˜π‘₯) = ((Οˆβ€˜π‘₯) βˆ’ π‘₯))
11410, 113syl 17 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘…β€˜π‘₯) = ((Οˆβ€˜π‘₯) βˆ’ π‘₯))
115114oveq1d 7416 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) = (((Οˆβ€˜π‘₯) βˆ’ π‘₯) Β· (logβ€˜π‘₯)))
11616recnd 11238 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Οˆβ€˜π‘₯) ∈ β„‚)
117116, 62, 12subdird 11667 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((Οˆβ€˜π‘₯) βˆ’ π‘₯) Β· (logβ€˜π‘₯)) = (((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))))
118115, 117eqtrd 2764 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) = (((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))))
119118oveq1d 7416 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) = ((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))))
12017recnd 11238 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) ∈ β„‚)
12162, 12mulcld 11230 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘₯ Β· (logβ€˜π‘₯)) ∈ β„‚)
122120, 60, 121addsubd 11588 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))) = ((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))))
123119, 122eqtr4d 2767 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) = ((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))))
124123oveq1d 7416 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) = (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))) / π‘₯))
12537recnd 11238 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) ∈ β„‚)
126125, 121, 62, 64divsubdird 12025 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))) / π‘₯) = (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ ((π‘₯ Β· (logβ€˜π‘₯)) / π‘₯)))
12712, 62, 64divcan3d 11991 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘₯ Β· (logβ€˜π‘₯)) / π‘₯) = (logβ€˜π‘₯))
128127oveq2d 7417 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ ((π‘₯ Β· (logβ€˜π‘₯)) / π‘₯)) = (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)))
129126, 128eqtrd 2764 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) βˆ’ (π‘₯ Β· (logβ€˜π‘₯))) / π‘₯) = (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)))
130124, 129eqtrd 2764 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) = (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)))
13152, 62, 64divcan3d 11991 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) / π‘₯) = ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))
132130, 131oveq12d 7419 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ ((π‘₯ Β· ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) / π‘₯)) = ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)) βˆ’ ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))))
13365, 112, 1323eqtr3rd 2773 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (logβ€˜π‘₯)) βˆ’ ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)))) = ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯))
13442, 53, 1333eqtrrd 2769 . . . 4 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) = ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))) βˆ’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯))))
135134mpteq2dva 5238 . . 3 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯)) = (π‘₯ ∈ (1(,)+∞) ↦ ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))) βˆ’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯)))))
13619, 11remulcld 11240 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (2 Β· (logβ€˜π‘₯)) ∈ ℝ)
13738, 136resubcld 11638 . . . 4 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))) ∈ ℝ)
13821, 50remulcld 11240 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) ∈ ℝ)
139138, 11resubcld 11638 . . . 4 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯)) ∈ ℝ)
140 selberg3 27396 . . . . 5 (π‘₯ ∈ (1(,)+∞) ↦ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯)))) ∈ 𝑂(1)
141140a1i 11 . . . 4 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ (((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯)))) ∈ 𝑂(1))
14219recnd 11238 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 2 ∈ β„‚)
14350, 20rerpdivcld 13043 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) ∈ ℝ)
144143recnd 11238 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) ∈ β„‚)
14511rehalfcld 12455 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((logβ€˜π‘₯) / 2) ∈ ℝ)
146145recnd 11238 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((logβ€˜π‘₯) / 2) ∈ β„‚)
147142, 144, 146subdid 11666 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (2 Β· ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) βˆ’ ((logβ€˜π‘₯) / 2))) = ((2 Β· (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯))) βˆ’ (2 Β· ((logβ€˜π‘₯) / 2))))
148142, 12, 51, 46div32d 12009 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) = (2 Β· (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯))))
149148eqcomd 2730 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (2 Β· (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯))) = ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))))
150 2ne0 12312 . . . . . . . . . 10 2 β‰  0
151150a1i 11 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 2 β‰  0)
15212, 142, 151divcan2d 11988 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (2 Β· ((logβ€˜π‘₯) / 2)) = (logβ€˜π‘₯))
153149, 152oveq12d 7419 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((2 Β· (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯))) βˆ’ (2 Β· ((logβ€˜π‘₯) / 2))) = (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯)))
154147, 153eqtrd 2764 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (2 Β· ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) βˆ’ ((logβ€˜π‘₯) / 2))) = (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯)))
155154mpteq2dva 5238 . . . . 5 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ (2 Β· ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) βˆ’ ((logβ€˜π‘₯) / 2)))) = (π‘₯ ∈ (1(,)+∞) ↦ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯))))
156143, 145resubcld 11638 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) βˆ’ ((logβ€˜π‘₯) / 2)) ∈ ℝ)
157 ioossre 13381 . . . . . . . 8 (1(,)+∞) βŠ† ℝ
158 o1const 15560 . . . . . . . 8 (((1(,)+∞) βŠ† ℝ ∧ 2 ∈ β„‚) β†’ (π‘₯ ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
159157, 44, 158mp2an 689 . . . . . . 7 (π‘₯ ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1)
160159a1i 11 . . . . . 6 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
161 vmalogdivsum 27376 . . . . . . 7 (π‘₯ ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) βˆ’ ((logβ€˜π‘₯) / 2))) ∈ 𝑂(1)
162161a1i 11 . . . . . 6 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) βˆ’ ((logβ€˜π‘₯) / 2))) ∈ 𝑂(1))
16319, 156, 160, 162o1mul2 15565 . . . . 5 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ (2 Β· ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›)) / (logβ€˜π‘₯)) βˆ’ ((logβ€˜π‘₯) / 2)))) ∈ 𝑂(1))
164155, 163eqeltrrd 2826 . . . 4 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯))) ∈ 𝑂(1))
165137, 139, 141, 164o1sub2 15566 . . 3 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ ((((((Οˆβ€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))) βˆ’ (((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) / 𝑛) Β· (logβ€˜π‘›))) βˆ’ (logβ€˜π‘₯)))) ∈ 𝑂(1))
166135, 165eqeltrd 2825 . 2 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯)) ∈ 𝑂(1))
167166mptru 1540 1 (π‘₯ ∈ (1(,)+∞) ↦ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) + ((2 / (logβ€˜π‘₯)) Β· Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (π‘…β€˜(π‘₯ / 𝑛))) Β· (logβ€˜π‘›)))) / π‘₯)) ∈ 𝑂(1)
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395   = wceq 1533  βŠ€wtru 1534   ∈ wcel 2098   β‰  wne 2932   βŠ† wss 3940   class class class wbr 5138   ↦ cmpt 5221  β€˜cfv 6533  (class class class)co 7401  β„‚cc 11103  β„cr 11104  0cc0 11105  1c1 11106   + caddc 11108   Β· cmul 11110  +∞cpnf 11241   < clt 11244   βˆ’ cmin 11440   / cdiv 11867  β„•cn 12208  2c2 12263  β„+crp 12970  (,)cioo 13320  ...cfz 13480  βŒŠcfl 13751  π‘‚(1)co1 15426  Ξ£csu 15628  logclog 26393  Ξ›cvma 26928  Οˆcchp 26929
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 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183  ax-addf 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-disj 5104  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-er 8698  df-map 8817  df-pm 8818  df-ixp 8887  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-fsupp 9357  df-fi 9401  df-sup 9432  df-inf 9433  df-oi 9500  df-dju 9891  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-xnn0 12541  df-z 12555  df-dec 12674  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ioc 13325  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-fac 14230  df-bc 14259  df-hash 14287  df-shft 15010  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-limsup 15411  df-clim 15428  df-rlim 15429  df-o1 15430  df-lo1 15431  df-sum 15629  df-ef 16007  df-e 16008  df-sin 16009  df-cos 16010  df-tan 16011  df-pi 16012  df-dvds 16194  df-gcd 16432  df-prm 16605  df-pc 16766  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-starv 17208  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-unif 17216  df-hom 17217  df-cco 17218  df-rest 17364  df-topn 17365  df-0g 17383  df-gsum 17384  df-topgen 17385  df-pt 17386  df-prds 17389  df-xrs 17444  df-qtop 17449  df-imas 17450  df-xps 17452  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18560  df-sgrp 18639  df-mnd 18655  df-submnd 18701  df-mulg 18983  df-cntz 19218  df-cmn 19687  df-psmet 21215  df-xmet 21216  df-met 21217  df-bl 21218  df-mopn 21219  df-fbas 21220  df-fg 21221  df-cnfld 21224  df-top 22706  df-topon 22723  df-topsp 22745  df-bases 22759  df-cld 22833  df-ntr 22834  df-cls 22835  df-nei 22912  df-lp 22950  df-perf 22951  df-cn 23041  df-cnp 23042  df-haus 23129  df-cmp 23201  df-tx 23376  df-hmeo 23569  df-fil 23660  df-fm 23752  df-flim 23753  df-flf 23754  df-xms 24136  df-ms 24137  df-tms 24138  df-cncf 24708  df-limc 25705  df-dv 25706  df-ulm 26218  df-log 26395  df-cxp 26396  df-atan 26703  df-em 26829  df-cht 26933  df-vma 26934  df-chp 26935  df-ppi 26936  df-mu 26937
This theorem is referenced by:  selberg34r  27408
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