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Theorem pntrlog2bndlem1 27069
Description: The sum of selberg3r 27061 and selberg4r 27062. (Contributed by Mario Carneiro, 31-May-2016.)
Hypotheses
Ref Expression
pntsval.1 𝑆 = (π‘Ž ∈ ℝ ↦ Σ𝑖 ∈ (1...(βŒŠβ€˜π‘Ž))((Ξ›β€˜π‘–) Β· ((logβ€˜π‘–) + (Οˆβ€˜(π‘Ž / 𝑖)))))
pntrlog2bnd.r 𝑅 = (π‘Ž ∈ ℝ+ ↦ ((Οˆβ€˜π‘Ž) βˆ’ π‘Ž))
Assertion
Ref Expression
pntrlog2bndlem1 (π‘₯ ∈ (1(,)+∞) ↦ ((((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) / π‘₯)) ∈ ≀𝑂(1)
Distinct variable groups:   𝑖,π‘Ž,𝑛,π‘₯   𝑆,𝑛,π‘₯   𝑅,𝑛,π‘₯
Allowed substitution hints:   𝑅(𝑖,π‘Ž)   𝑆(𝑖,π‘Ž)

Proof of Theorem pntrlog2bndlem1
Dummy variables π‘˜ π‘š 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1red 11211 . . 3 (⊀ β†’ 1 ∈ ℝ)
2 pntrlog2bnd.r . . . . 5 𝑅 = (π‘Ž ∈ ℝ+ ↦ ((Οˆβ€˜π‘Ž) βˆ’ π‘Ž))
32selberg34r 27063 . . . 4 (π‘₯ ∈ (1(,)+∞) ↦ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯)) ∈ 𝑂(1)
4 elioore 13350 . . . . . . . . . . . 12 (π‘₯ ∈ (1(,)+∞) β†’ π‘₯ ∈ ℝ)
54adantl 482 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ ℝ)
6 1rp 12974 . . . . . . . . . . . 12 1 ∈ ℝ+
76a1i 11 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ∈ ℝ+)
8 1red 11211 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ∈ ℝ)
9 eliooord 13379 . . . . . . . . . . . . . 14 (π‘₯ ∈ (1(,)+∞) β†’ (1 < π‘₯ ∧ π‘₯ < +∞))
109adantl 482 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (1 < π‘₯ ∧ π‘₯ < +∞))
1110simpld 495 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 < π‘₯)
128, 5, 11ltled 11358 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ≀ π‘₯)
135, 7, 12rpgecld 13051 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ ℝ+)
142pntrf 27055 . . . . . . . . . . 11 𝑅:ℝ+βŸΆβ„
1514ffvelcdmi 7082 . . . . . . . . . 10 (π‘₯ ∈ ℝ+ β†’ (π‘…β€˜π‘₯) ∈ ℝ)
1613, 15syl 17 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘…β€˜π‘₯) ∈ ℝ)
1713relogcld 26122 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ ℝ)
1816, 17remulcld 11240 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) ∈ ℝ)
19 fzfid 13934 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (1...(βŒŠβ€˜π‘₯)) ∈ Fin)
2013adantr 481 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ π‘₯ ∈ ℝ+)
21 elfznn 13526 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ β„•)
2221adantl 482 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ β„•)
2322nnrpd 13010 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ ℝ+)
2420, 23rpdivcld 13029 . . . . . . . . . . . 12 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ ℝ+)
2514ffvelcdmi 7082 . . . . . . . . . . . 12 ((π‘₯ / 𝑛) ∈ ℝ+ β†’ (π‘…β€˜(π‘₯ / 𝑛)) ∈ ℝ)
2624, 25syl 17 . . . . . . . . . . 11 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘…β€˜(π‘₯ / 𝑛)) ∈ ℝ)
27 fzfid 13934 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...𝑛) ∈ Fin)
28 dvdsssfz1 16257 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} βŠ† (1...𝑛))
2922, 28syl 17 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} βŠ† (1...𝑛))
3027, 29ssfid 9263 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ∈ Fin)
31 ssrab2 4076 . . . . . . . . . . . . . . . 16 {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} βŠ† β„•
32 simpr 485 . . . . . . . . . . . . . . . 16 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
3331, 32sselid 3979 . . . . . . . . . . . . . . 15 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ π‘š ∈ β„•)
34 vmacl 26611 . . . . . . . . . . . . . . 15 (π‘š ∈ β„• β†’ (Ξ›β€˜π‘š) ∈ ℝ)
3533, 34syl 17 . . . . . . . . . . . . . 14 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (Ξ›β€˜π‘š) ∈ ℝ)
36 dvdsdivcl 16255 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ β„• ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / π‘š) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
3722, 36sylan 580 . . . . . . . . . . . . . . . 16 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / π‘š) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
3831, 37sselid 3979 . . . . . . . . . . . . . . 15 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / π‘š) ∈ β„•)
39 vmacl 26611 . . . . . . . . . . . . . . 15 ((𝑛 / π‘š) ∈ β„• β†’ (Ξ›β€˜(𝑛 / π‘š)) ∈ ℝ)
4038, 39syl 17 . . . . . . . . . . . . . 14 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (Ξ›β€˜(𝑛 / π‘š)) ∈ ℝ)
4135, 40remulcld 11240 . . . . . . . . . . . . 13 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) ∈ ℝ)
4230, 41fsumrecl 15676 . . . . . . . . . . . 12 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) ∈ ℝ)
43 vmacl 26611 . . . . . . . . . . . . . 14 (𝑛 ∈ β„• β†’ (Ξ›β€˜π‘›) ∈ ℝ)
4422, 43syl 17 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘›) ∈ ℝ)
4523relogcld 26122 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (logβ€˜π‘›) ∈ ℝ)
4644, 45remulcld 11240 . . . . . . . . . . . 12 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) ∈ ℝ)
4742, 46resubcld 11638 . . . . . . . . . . 11 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) ∈ ℝ)
4826, 47remulcld 11240 . . . . . . . . . 10 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) ∈ ℝ)
4919, 48fsumrecl 15676 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) ∈ ℝ)
505, 11rplogcld 26128 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ ℝ+)
5149, 50rerpdivcld 13043 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)) ∈ ℝ)
5218, 51resubcld 11638 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) ∈ ℝ)
5352, 13rerpdivcld 13043 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯) ∈ ℝ)
5453recnd 11238 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯) ∈ β„‚)
5554lo1o12 15473 . . . 4 (⊀ β†’ ((π‘₯ ∈ (1(,)+∞) ↦ ((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯)) ∈ 𝑂(1) ↔ (π‘₯ ∈ (1(,)+∞) ↦ (absβ€˜((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯))) ∈ ≀𝑂(1)))
563, 55mpbii 232 . . 3 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ (absβ€˜((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯))) ∈ ≀𝑂(1))
5754abscld 15379 . . 3 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯)) ∈ ℝ)
5816recnd 11238 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘…β€˜π‘₯) ∈ β„‚)
5958abscld 15379 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜(π‘…β€˜π‘₯)) ∈ ℝ)
6059, 17remulcld 11240 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) ∈ ℝ)
6126recnd 11238 . . . . . . . . 9 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘…β€˜(π‘₯ / 𝑛)) ∈ β„‚)
6261abscld 15379 . . . . . . . 8 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) ∈ ℝ)
6322nnred 12223 . . . . . . . . . 10 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ ℝ)
64 pntsval.1 . . . . . . . . . . . 12 𝑆 = (π‘Ž ∈ ℝ ↦ Σ𝑖 ∈ (1...(βŒŠβ€˜π‘Ž))((Ξ›β€˜π‘–) Β· ((logβ€˜π‘–) + (Οˆβ€˜(π‘Ž / 𝑖)))))
6564pntsf 27065 . . . . . . . . . . 11 𝑆:β„βŸΆβ„
6665ffvelcdmi 7082 . . . . . . . . . 10 (𝑛 ∈ ℝ β†’ (π‘†β€˜π‘›) ∈ ℝ)
6763, 66syl 17 . . . . . . . . 9 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜π‘›) ∈ ℝ)
68 1red 11211 . . . . . . . . . . 11 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 1 ∈ ℝ)
6963, 68resubcld 11638 . . . . . . . . . 10 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (𝑛 βˆ’ 1) ∈ ℝ)
7065ffvelcdmi 7082 . . . . . . . . . 10 ((𝑛 βˆ’ 1) ∈ ℝ β†’ (π‘†β€˜(𝑛 βˆ’ 1)) ∈ ℝ)
7169, 70syl 17 . . . . . . . . 9 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜(𝑛 βˆ’ 1)) ∈ ℝ)
7267, 71resubcld 11638 . . . . . . . 8 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1))) ∈ ℝ)
7362, 72remulcld 11240 . . . . . . 7 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) ∈ ℝ)
7419, 73fsumrecl 15676 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) ∈ ℝ)
7574, 50rerpdivcld 13043 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯)) ∈ ℝ)
7660, 75resubcld 11638 . . . 4 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) ∈ ℝ)
7776, 13rerpdivcld 13043 . . 3 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) / π‘₯) ∈ ℝ)
7817recnd 11238 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ β„‚)
7958, 78mulcld 11230 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) ∈ β„‚)
8049recnd 11238 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) ∈ β„‚)
8150rpne0d 13017 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) β‰  0)
8280, 78, 81divcld 11986 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)) ∈ β„‚)
8379, 82subcld 11567 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) ∈ β„‚)
8483abscld 15379 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))) ∈ ℝ)
8580abscld 15379 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) ∈ ℝ)
8685, 50rerpdivcld 13043 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (logβ€˜π‘₯)) ∈ ℝ)
8760, 86resubcld 11638 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (logβ€˜π‘₯))) ∈ ℝ)
8848recnd 11238 . . . . . . . . . . . 12 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) ∈ β„‚)
8988abscld 15379 . . . . . . . . . . 11 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) ∈ ℝ)
9019, 89fsumrecl 15676 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(absβ€˜((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) ∈ ℝ)
9119, 88fsumabs 15743 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) ≀ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(absβ€˜((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))))
9247recnd 11238 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) ∈ β„‚)
9361, 92absmuld 15397 . . . . . . . . . . . 12 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) = ((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· (absβ€˜(Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))))
9492abscld 15379 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜(Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) ∈ ℝ)
9561absge0d 15387 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ (absβ€˜(π‘…β€˜(π‘₯ / 𝑛))))
9642recnd 11238 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) ∈ β„‚)
9746recnd 11238 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) ∈ β„‚)
9896, 97abs2dif2d 15401 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜(Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) ≀ ((absβ€˜Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š)))) + (absβ€˜((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))))
9971recnd 11238 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜(𝑛 βˆ’ 1)) ∈ β„‚)
10096, 97addcld 11229 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) ∈ β„‚)
10199, 100pncan2d 11569 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (((π‘†β€˜(𝑛 βˆ’ 1)) + (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1))) = (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
102 elfzuz 13493 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ (β„€β‰₯β€˜1))
103102adantl 482 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ (β„€β‰₯β€˜1))
104 elfznn 13526 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘˜ ∈ (1...𝑛) β†’ π‘˜ ∈ β„•)
105104adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ π‘˜ ∈ β„•)
106 vmacl 26611 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ ∈ β„• β†’ (Ξ›β€˜π‘˜) ∈ ℝ)
107105, 106syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ (Ξ›β€˜π‘˜) ∈ ℝ)
108105nnrpd 13010 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ π‘˜ ∈ ℝ+)
109108relogcld 26122 . . . . . . . . . . . . . . . . . . . . 21 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ (logβ€˜π‘˜) ∈ ℝ)
110107, 109remulcld 11240 . . . . . . . . . . . . . . . . . . . 20 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ ((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) ∈ ℝ)
111 fzfid 13934 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ (1...π‘˜) ∈ Fin)
112 dvdsssfz1 16257 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘˜ ∈ β„• β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} βŠ† (1...π‘˜))
113105, 112syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} βŠ† (1...π‘˜))
114111, 113ssfid 9263 . . . . . . . . . . . . . . . . . . . . 21 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ∈ Fin)
115 ssrab2 4076 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} βŠ† β„•
116 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜})
117115, 116sselid 3979 . . . . . . . . . . . . . . . . . . . . . . 23 (((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ π‘š ∈ β„•)
118117, 34syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ (Ξ›β€˜π‘š) ∈ ℝ)
119 dvdsdivcl 16255 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((π‘˜ ∈ β„• ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ (π‘˜ / π‘š) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜})
120105, 119sylan 580 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ (π‘˜ / π‘š) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜})
121115, 120sselid 3979 . . . . . . . . . . . . . . . . . . . . . . 23 (((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ (π‘˜ / π‘š) ∈ β„•)
122 vmacl 26611 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘˜ / π‘š) ∈ β„• β†’ (Ξ›β€˜(π‘˜ / π‘š)) ∈ ℝ)
123121, 122syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ (Ξ›β€˜(π‘˜ / π‘š)) ∈ ℝ)
124118, 123remulcld 11240 . . . . . . . . . . . . . . . . . . . . 21 (((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š))) ∈ ℝ)
125114, 124fsumrecl 15676 . . . . . . . . . . . . . . . . . . . 20 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š))) ∈ ℝ)
126110, 125readdcld 11239 . . . . . . . . . . . . . . . . . . 19 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ (((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))) ∈ ℝ)
127126recnd 11238 . . . . . . . . . . . . . . . . . 18 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ (((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))) ∈ β„‚)
128 fveq2 6888 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑛 β†’ (Ξ›β€˜π‘˜) = (Ξ›β€˜π‘›))
129 fveq2 6888 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑛 β†’ (logβ€˜π‘˜) = (logβ€˜π‘›))
130128, 129oveq12d 7423 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 𝑛 β†’ ((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) = ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))
131 breq2 5151 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = 𝑛 β†’ (𝑦 βˆ₯ π‘˜ ↔ 𝑦 βˆ₯ 𝑛))
132131rabbidv 3440 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑛 β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} = {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
133 fvoveq1 7428 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ = 𝑛 β†’ (Ξ›β€˜(π‘˜ / π‘š)) = (Ξ›β€˜(𝑛 / π‘š)))
134133oveq2d 7421 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = 𝑛 β†’ ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š))) = ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))))
135134adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((π‘˜ = 𝑛 ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š))) = ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))))
136132, 135sumeq12rdv 15649 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 𝑛 β†’ Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š))) = Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))))
137130, 136oveq12d 7423 . . . . . . . . . . . . . . . . . 18 (π‘˜ = 𝑛 β†’ (((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))) = (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š)))))
138103, 127, 137fsumm1 15693 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Ξ£π‘˜ ∈ (1...𝑛)(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))) = (Ξ£π‘˜ ∈ (1...(𝑛 βˆ’ 1))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))) + (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))))))
13964pntsval2 27068 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ β†’ (π‘†β€˜π‘›) = Ξ£π‘˜ ∈ (1...(βŒŠβ€˜π‘›))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))))
14063, 139syl 17 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜π‘›) = Ξ£π‘˜ ∈ (1...(βŒŠβ€˜π‘›))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))))
14122nnzd 12581 . . . . . . . . . . . . . . . . . . . . 21 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ β„€)
142 flid 13769 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ β„€ β†’ (βŒŠβ€˜π‘›) = 𝑛)
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . 20 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (βŒŠβ€˜π‘›) = 𝑛)
144143oveq2d 7421 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...(βŒŠβ€˜π‘›)) = (1...𝑛))
145144sumeq1d 15643 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Ξ£π‘˜ ∈ (1...(βŒŠβ€˜π‘›))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))) = Ξ£π‘˜ ∈ (1...𝑛)(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))))
146140, 145eqtrd 2772 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜π‘›) = Ξ£π‘˜ ∈ (1...𝑛)(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))))
14764pntsval2 27068 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 βˆ’ 1) ∈ ℝ β†’ (π‘†β€˜(𝑛 βˆ’ 1)) = Ξ£π‘˜ ∈ (1...(βŒŠβ€˜(𝑛 βˆ’ 1)))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))))
14869, 147syl 17 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜(𝑛 βˆ’ 1)) = Ξ£π‘˜ ∈ (1...(βŒŠβ€˜(𝑛 βˆ’ 1)))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))))
149 1zzd 12589 . . . . . . . . . . . . . . . . . . . . . . 23 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 1 ∈ β„€)
150141, 149zsubcld 12667 . . . . . . . . . . . . . . . . . . . . . 22 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (𝑛 βˆ’ 1) ∈ β„€)
151 flid 13769 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 βˆ’ 1) ∈ β„€ β†’ (βŒŠβ€˜(𝑛 βˆ’ 1)) = (𝑛 βˆ’ 1))
152150, 151syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (βŒŠβ€˜(𝑛 βˆ’ 1)) = (𝑛 βˆ’ 1))
153152oveq2d 7421 . . . . . . . . . . . . . . . . . . . 20 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...(βŒŠβ€˜(𝑛 βˆ’ 1))) = (1...(𝑛 βˆ’ 1)))
154153sumeq1d 15643 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Ξ£π‘˜ ∈ (1...(βŒŠβ€˜(𝑛 βˆ’ 1)))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))) = Ξ£π‘˜ ∈ (1...(𝑛 βˆ’ 1))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))))
155148, 154eqtrd 2772 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜(𝑛 βˆ’ 1)) = Ξ£π‘˜ ∈ (1...(𝑛 βˆ’ 1))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))))
15696, 97addcomd 11412 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) = (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š)))))
157155, 156oveq12d 7423 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((π‘†β€˜(𝑛 βˆ’ 1)) + (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) = (Ξ£π‘˜ ∈ (1...(𝑛 βˆ’ 1))(((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(π‘˜ / π‘š)))) + (((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) + Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))))))
158138, 146, 1573eqtr4d 2782 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜π‘›) = ((π‘†β€˜(𝑛 βˆ’ 1)) + (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))))
159158oveq1d 7420 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1))) = (((π‘†β€˜(𝑛 βˆ’ 1)) + (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1))))
160 vmage0 26614 . . . . . . . . . . . . . . . . . . . 20 (π‘š ∈ β„• β†’ 0 ≀ (Ξ›β€˜π‘š))
16133, 160syl 17 . . . . . . . . . . . . . . . . . . 19 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ 0 ≀ (Ξ›β€˜π‘š))
162 vmage0 26614 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 / π‘š) ∈ β„• β†’ 0 ≀ (Ξ›β€˜(𝑛 / π‘š)))
16338, 162syl 17 . . . . . . . . . . . . . . . . . . 19 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ 0 ≀ (Ξ›β€˜(𝑛 / π‘š)))
16435, 40, 161, 163mulge0d 11787 . . . . . . . . . . . . . . . . . 18 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ 0 ≀ ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))))
16530, 41, 164fsumge0 15737 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))))
16642, 165absidd 15365 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š)))) = Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))))
167 vmage0 26614 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ β„• β†’ 0 ≀ (Ξ›β€˜π‘›))
16822, 167syl 17 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ (Ξ›β€˜π‘›))
16922nnge1d 12256 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 1 ≀ 𝑛)
17063, 169logge0d 26129 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ (logβ€˜π‘›))
17144, 45, 168, 170mulge0d 11787 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))
17246, 171absidd 15365 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) = ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))
173166, 172oveq12d 7423 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((absβ€˜Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š)))) + (absβ€˜((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) = (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
174101, 159, 1733eqtr4d 2782 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1))) = ((absβ€˜Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š)))) + (absβ€˜((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))))
17598, 174breqtrrd 5175 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜(Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) ≀ ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1))))
17694, 72, 62, 95, 175lemul2ad 12150 . . . . . . . . . . . 12 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· (absβ€˜(Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) ≀ ((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))))
17793, 176eqbrtrd 5169 . . . . . . . . . . 11 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) ≀ ((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))))
17819, 89, 73, 177fsumle 15741 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(absβ€˜((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) ≀ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))))
17985, 90, 74, 91, 178letrd 11367 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) ≀ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))))
18085, 74, 50, 179lediv1dd 13070 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (logβ€˜π‘₯)) ≀ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯)))
18186, 75, 60, 180lesub2dd 11827 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) ≀ (((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (logβ€˜π‘₯))))
18258, 78absmuld 15397 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯))) = ((absβ€˜(π‘…β€˜π‘₯)) Β· (absβ€˜(logβ€˜π‘₯))))
1835, 12logge0d 26129 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 0 ≀ (logβ€˜π‘₯))
18417, 183absidd 15365 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜(logβ€˜π‘₯)) = (logβ€˜π‘₯))
185184oveq2d 7421 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜(π‘…β€˜π‘₯)) Β· (absβ€˜(logβ€˜π‘₯))) = ((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)))
186182, 185eqtrd 2772 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯))) = ((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)))
18780, 78, 81absdivd 15398 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜(Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) = ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (absβ€˜(logβ€˜π‘₯))))
188184oveq2d 7421 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (absβ€˜(logβ€˜π‘₯))) = ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (logβ€˜π‘₯)))
189187, 188eqtrd 2772 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜(Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) = ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (logβ€˜π‘₯)))
190186, 189oveq12d 7423 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯))) βˆ’ (absβ€˜(Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))) = (((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (logβ€˜π‘₯))))
19179, 82abs2difd 15400 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯))) βˆ’ (absβ€˜(Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))) ≀ (absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))))
192190, 191eqbrtrrd 5171 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ ((absβ€˜Ξ£π‘› ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))) / (logβ€˜π‘₯))) ≀ (absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))))
19376, 87, 84, 181, 192letrd 11367 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) ≀ (absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))))
19476, 84, 13, 193lediv1dd 13070 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) / π‘₯) ≀ ((absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))) / π‘₯))
19552recnd 11238 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) ∈ β„‚)
1965recnd 11238 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ β„‚)
19713rpne0d 13017 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ β‰  0)
198195, 196, 197absdivd 15398 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯)) = ((absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))) / (absβ€˜π‘₯)))
19913rpge0d 13016 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 0 ≀ π‘₯)
2005, 199absidd 15365 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜π‘₯) = π‘₯)
201200oveq2d 7421 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))) / (absβ€˜π‘₯)) = ((absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))) / π‘₯))
202198, 201eqtrd 2772 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯)) = ((absβ€˜(((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯)))) / π‘₯))
203194, 202breqtrrd 5175 . . . 4 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) / π‘₯) ≀ (absβ€˜((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯)))
204203adantrr 715 . . 3 ((⊀ ∧ (π‘₯ ∈ (1(,)+∞) ∧ 1 ≀ π‘₯)) β†’ ((((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) / π‘₯) ≀ (absβ€˜((((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((π‘…β€˜(π‘₯ / 𝑛)) Β· (Ξ£π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘š) Β· (Ξ›β€˜(𝑛 / π‘š))) βˆ’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))) / (logβ€˜π‘₯))) / π‘₯)))
2051, 56, 57, 77, 204lo1le 15594 . 2 (⊀ β†’ (π‘₯ ∈ (1(,)+∞) ↦ ((((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) / π‘₯)) ∈ ≀𝑂(1))
206205mptru 1548 1 (π‘₯ ∈ (1(,)+∞) ↦ ((((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) βˆ’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) Β· ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1)))) / (logβ€˜π‘₯))) / π‘₯)) ∈ ≀𝑂(1)
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 396   = wceq 1541  βŠ€wtru 1542   ∈ wcel 2106  {crab 3432   βŠ† wss 3947   class class class wbr 5147   ↦ cmpt 5230  β€˜cfv 6540  (class class class)co 7405  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   Β· cmul 11111  +∞cpnf 11241   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440   / cdiv 11867  β„•cn 12208  β„€cz 12554  β„€β‰₯cuz 12818  β„+crp 12970  (,)cioo 13320  ...cfz 13480  βŒŠcfl 13751  abscabs 15177  π‘‚(1)co1 15426  β‰€π‘‚(1)clo1 15427  Ξ£csu 15628   βˆ₯ cdvds 16193  logclog 26054  Ξ›cvma 26585  Οˆcchp 26586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  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 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-mulg 18945  df-cntz 19175  df-cmn 19644  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-fbas 20933  df-fg 20934  df-cnfld 20937  df-top 22387  df-topon 22404  df-topsp 22426  df-bases 22440  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-lp 22631  df-perf 22632  df-cn 22722  df-cnp 22723  df-haus 22810  df-cmp 22882  df-tx 23057  df-hmeo 23250  df-fil 23341  df-fm 23433  df-flim 23434  df-flf 23435  df-xms 23817  df-ms 23818  df-tms 23819  df-cncf 24385  df-limc 25374  df-dv 25375  df-ulm 25880  df-log 26056  df-cxp 26057  df-atan 26361  df-em 26486  df-cht 26590  df-vma 26591  df-chp 26592  df-ppi 26593  df-mu 26594
This theorem is referenced by:  pntrlog2bndlem4  27072
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