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Theorem pntrlog2bndlem1 26941
Description: The sum of selberg3r 26933 and selberg4r 26934. (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 11163 . . 3 (⊀ β†’ 1 ∈ ℝ)
2 pntrlog2bnd.r . . . . 5 𝑅 = (π‘Ž ∈ ℝ+ ↦ ((Οˆβ€˜π‘Ž) βˆ’ π‘Ž))
32selberg34r 26935 . . . 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 13301 . . . . . . . . . . . 12 (π‘₯ ∈ (1(,)+∞) β†’ π‘₯ ∈ ℝ)
54adantl 483 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ ℝ)
6 1rp 12926 . . . . . . . . . . . 12 1 ∈ ℝ+
76a1i 11 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ∈ ℝ+)
8 1red 11163 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ∈ ℝ)
9 eliooord 13330 . . . . . . . . . . . . . 14 (π‘₯ ∈ (1(,)+∞) β†’ (1 < π‘₯ ∧ π‘₯ < +∞))
109adantl 483 . . . . . . . . . . . . 13 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (1 < π‘₯ ∧ π‘₯ < +∞))
1110simpld 496 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 < π‘₯)
128, 5, 11ltled 11310 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 1 ≀ π‘₯)
135, 7, 12rpgecld 13003 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ ℝ+)
142pntrf 26927 . . . . . . . . . . 11 𝑅:ℝ+βŸΆβ„
1514ffvelcdmi 7039 . . . . . . . . . 10 (π‘₯ ∈ ℝ+ β†’ (π‘…β€˜π‘₯) ∈ ℝ)
1613, 15syl 17 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘…β€˜π‘₯) ∈ ℝ)
1713relogcld 25994 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ ℝ)
1816, 17remulcld 11192 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) ∈ ℝ)
19 fzfid 13885 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (1...(βŒŠβ€˜π‘₯)) ∈ Fin)
2013adantr 482 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ π‘₯ ∈ ℝ+)
21 elfznn 13477 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ β„•)
2221adantl 483 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ β„•)
2322nnrpd 12962 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ ℝ+)
2420, 23rpdivcld 12981 . . . . . . . . . . . 12 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ ℝ+)
2514ffvelcdmi 7039 . . . . . . . . . . . 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 13885 . . . . . . . . . . . . . 14 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...𝑛) ∈ Fin)
28 dvdsssfz1 16207 . . . . . . . . . . . . . . 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 9218 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ∈ Fin)
31 ssrab2 4042 . . . . . . . . . . . . . . . 16 {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} βŠ† β„•
32 simpr 486 . . . . . . . . . . . . . . . 16 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
3331, 32sselid 3947 . . . . . . . . . . . . . . 15 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ π‘š ∈ β„•)
34 vmacl 26483 . . . . . . . . . . . . . . 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 16205 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ β„• ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / π‘š) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
3722, 36sylan 581 . . . . . . . . . . . . . . . 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 3947 . . . . . . . . . . . . . . 15 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / π‘š) ∈ β„•)
39 vmacl 26483 . . . . . . . . . . . . . . 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 11192 . . . . . . . . . . . . 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 15626 . . . . . . . . . . . 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 26483 . . . . . . . . . . . . . 14 (𝑛 ∈ β„• β†’ (Ξ›β€˜π‘›) ∈ ℝ)
4422, 43syl 17 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘›) ∈ ℝ)
4523relogcld 25994 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (logβ€˜π‘›) ∈ ℝ)
4644, 45remulcld 11192 . . . . . . . . . . . 12 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) ∈ ℝ)
4742, 46resubcld 11590 . . . . . . . . . . 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 11192 . . . . . . . . . 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 15626 . . . . . . . . 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 26000 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ ℝ+)
5149, 50rerpdivcld 12995 . . . . . . . 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 11590 . . . . . . 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 12995 . . . . . 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 11190 . . . . 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 15422 . . . 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 15328 . . 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 11190 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (π‘…β€˜π‘₯) ∈ β„‚)
5958abscld 15328 . . . . . 6 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜(π‘…β€˜π‘₯)) ∈ ℝ)
6059, 17remulcld 11192 . . . . 5 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)) ∈ ℝ)
6126recnd 11190 . . . . . . . . 9 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘…β€˜(π‘₯ / 𝑛)) ∈ β„‚)
6261abscld 15328 . . . . . . . 8 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜(π‘…β€˜(π‘₯ / 𝑛))) ∈ ℝ)
6322nnred 12175 . . . . . . . . . 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 26937 . . . . . . . . . . 11 𝑆:β„βŸΆβ„
6665ffvelcdmi 7039 . . . . . . . . . 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 11163 . . . . . . . . . . 11 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 1 ∈ ℝ)
6963, 68resubcld 11590 . . . . . . . . . 10 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (𝑛 βˆ’ 1) ∈ ℝ)
7065ffvelcdmi 7039 . . . . . . . . . 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 11590 . . . . . . . 8 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((π‘†β€˜π‘›) βˆ’ (π‘†β€˜(𝑛 βˆ’ 1))) ∈ ℝ)
7362, 72remulcld 11192 . . . . . . 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 15626 . . . . . 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 12995 . . . . 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 11590 . . . 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 12995 . . 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 11190 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) ∈ β„‚)
7958, 78mulcld 11182 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯)) ∈ β„‚)
8049recnd 11190 . . . . . . . . 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 12969 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (logβ€˜π‘₯) β‰  0)
8280, 78, 81divcld 11938 . . . . . . . 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 11519 . . . . . . 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 15328 . . . . . 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 15328 . . . . . . . . 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 12995 . . . . . . . 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 11590 . . . . . . 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 11190 . . . . . . . . . . . 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 15328 . . . . . . . . . . 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 15626 . . . . . . . . . 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 15693 . . . . . . . . . 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 11190 . . . . . . . . . . . . 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 15346 . . . . . . . . . . . 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 15328 . . . . . . . . . . . . 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 15336 . . . . . . . . . . . . 13 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ (absβ€˜(π‘…β€˜(π‘₯ / 𝑛))))
9642recnd 11190 . . . . . . . . . . . . . . 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 11190 . . . . . . . . . . . . . . 15 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) ∈ β„‚)
9896, 97abs2dif2d 15350 . . . . . . . . . . . . . 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 11190 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘†β€˜(𝑛 βˆ’ 1)) ∈ β„‚)
10096, 97addcld 11181 . . . . . . . . . . . . . . . 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 11521 . . . . . . . . . . . . . . 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 13444 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ (β„€β‰₯β€˜1))
103102adantl 483 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ (β„€β‰₯β€˜1))
104 elfznn 13477 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘˜ ∈ (1...𝑛) β†’ π‘˜ ∈ β„•)
105104adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ π‘˜ ∈ β„•)
106 vmacl 26483 . . . . . . . . . . . . . . . . . . . . . 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 12962 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ π‘˜ ∈ ℝ+)
109108relogcld 25994 . . . . . . . . . . . . . . . . . . . . 21 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ (logβ€˜π‘˜) ∈ ℝ)
110107, 109remulcld 11192 . . . . . . . . . . . . . . . . . . . 20 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ ((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) ∈ ℝ)
111 fzfid 13885 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ (1...π‘˜) ∈ Fin)
112 dvdsssfz1 16207 . . . . . . . . . . . . . . . . . . . . . . 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 9218 . . . . . . . . . . . . . . . . . . . . 21 ((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} ∈ Fin)
115 ssrab2 4042 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} βŠ† β„•
116 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . 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 3947 . . . . . . . . . . . . . . . . . . . . . . 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 16205 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((π‘˜ ∈ β„• ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ (π‘˜ / π‘š) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜})
120105, 119sylan 581 . . . . . . . . . . . . . . . . . . . . . . . 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 3947 . . . . . . . . . . . . . . . . . . . . . . 23 (((((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘˜ ∈ (1...𝑛)) ∧ π‘š ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜}) β†’ (π‘˜ / π‘š) ∈ β„•)
122 vmacl 26483 . . . . . . . . . . . . . . . . . . . . . . 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 11192 . . . . . . . . . . . . . . . . . . . . 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 15626 . . . . . . . . . . . . . . . . . . . 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 11191 . . . . . . . . . . . . . . . . . . 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 11190 . . . . . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑛 β†’ (Ξ›β€˜π‘˜) = (Ξ›β€˜π‘›))
129 fveq2 6847 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑛 β†’ (logβ€˜π‘˜) = (logβ€˜π‘›))
130128, 129oveq12d 7380 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 𝑛 β†’ ((Ξ›β€˜π‘˜) Β· (logβ€˜π‘˜)) = ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))
131 breq2 5114 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = 𝑛 β†’ (𝑦 βˆ₯ π‘˜ ↔ 𝑦 βˆ₯ 𝑛))
132131rabbidv 3418 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑛 β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ π‘˜} = {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
133 fvoveq1 7385 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ = 𝑛 β†’ (Ξ›β€˜(π‘˜ / π‘š)) = (Ξ›β€˜(𝑛 / π‘š)))
134133oveq2d 7378 . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . 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 15599 . . . . . . . . . . . . . . . . . . 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 7380 . . . . . . . . . . . . . . . . . 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 15643 . . . . . . . . . . . . . . . . 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 26940 . . . . . . . . . . . . . . . . . . 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 12533 . . . . . . . . . . . . . . . . . . . . 21 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ β„€)
142 flid 13720 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ β„€ β†’ (βŒŠβ€˜π‘›) = 𝑛)
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . 20 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (βŒŠβ€˜π‘›) = 𝑛)
144143oveq2d 7378 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...(βŒŠβ€˜π‘›)) = (1...𝑛))
145144sumeq1d 15593 . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . 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 26940 . . . . . . . . . . . . . . . . . . . 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 12541 . . . . . . . . . . . . . . . . . . . . . . 23 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 1 ∈ β„€)
150141, 149zsubcld 12619 . . . . . . . . . . . . . . . . . . . . . 22 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (𝑛 βˆ’ 1) ∈ β„€)
151 flid 13720 . . . . . . . . . . . . . . . . . . . . . 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 7378 . . . . . . . . . . . . . . . . . . . 20 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...(βŒŠβ€˜(𝑛 βˆ’ 1))) = (1...(𝑛 βˆ’ 1)))
154153sumeq1d 15593 . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . . 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 11364 . . . . . . . . . . . . . . . . . 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 7380 . . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . 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 7377 . . . . . . . . . . . . . . 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 26486 . . . . . . . . . . . . . . . . . . . 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 26486 . . . . . . . . . . . . . . . . . . . 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 11739 . . . . . . . . . . . . . . . . . 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 15687 . . . . . . . . . . . . . . . . 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 15314 . . . . . . . . . . . . . . . 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 26486 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ β„• β†’ 0 ≀ (Ξ›β€˜π‘›))
16822, 167syl 17 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ (Ξ›β€˜π‘›))
16922nnge1d 12208 . . . . . . . . . . . . . . . . . . 19 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 1 ≀ 𝑛)
17063, 169logge0d 26001 . . . . . . . . . . . . . . . . . 18 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ (logβ€˜π‘›))
17144, 45, 168, 170mulge0d 11739 . . . . . . . . . . . . . . . . 17 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 0 ≀ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))
17246, 171absidd 15314 . . . . . . . . . . . . . . . 16 (((⊀ ∧ π‘₯ ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (absβ€˜((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) = ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)))
173166, 172oveq12d 7380 . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . 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 5138 . . . . . . . . . . . . 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 12102 . . . . . . . . . . . 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 5132 . . . . . . . . . . 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 15691 . . . . . . . . . 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 11319 . . . . . . . . 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 13022 . . . . . . . 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 11779 . . . . . . 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 15346 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯))) = ((absβ€˜(π‘…β€˜π‘₯)) Β· (absβ€˜(logβ€˜π‘₯))))
1835, 12logge0d 26001 . . . . . . . . . . . 12 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 0 ≀ (logβ€˜π‘₯))
18417, 183absidd 15314 . . . . . . . . . . 11 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜(logβ€˜π‘₯)) = (logβ€˜π‘₯))
185184oveq2d 7378 . . . . . . . . . 10 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ ((absβ€˜(π‘…β€˜π‘₯)) Β· (absβ€˜(logβ€˜π‘₯))) = ((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)))
186182, 185eqtrd 2777 . . . . . . . . 9 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜((π‘…β€˜π‘₯) Β· (logβ€˜π‘₯))) = ((absβ€˜(π‘…β€˜π‘₯)) Β· (logβ€˜π‘₯)))
18780, 78, 81absdivd 15347 . . . . . . . . . 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 7378 . . . . . . . . . 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 2777 . . . . . . . . 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 7380 . . . . . . . 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 15349 . . . . . . . 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 5134 . . . . . . 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 11319 . . . . . 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 13022 . . . . 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 11190 . . . . . . 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 11190 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ ∈ β„‚)
19713rpne0d 12969 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ π‘₯ β‰  0)
198195, 196, 197absdivd 15347 . . . . . 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 12968 . . . . . . . 8 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ 0 ≀ π‘₯)
2005, 199absidd 15314 . . . . . . 7 ((⊀ ∧ π‘₯ ∈ (1(,)+∞)) β†’ (absβ€˜π‘₯) = π‘₯)
201200oveq2d 7378 . . . . . 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 2777 . . . . 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 5138 . . . 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 716 . . 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 15543 . 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 1549 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 397   = wceq 1542  βŠ€wtru 1543   ∈ wcel 2107  {crab 3410   βŠ† wss 3915   class class class wbr 5110   ↦ cmpt 5193  β€˜cfv 6501  (class class class)co 7362  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   Β· cmul 11063  +∞cpnf 11193   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392   / cdiv 11819  β„•cn 12160  β„€cz 12506  β„€β‰₯cuz 12770  β„+crp 12922  (,)cioo 13271  ...cfz 13431  βŒŠcfl 13702  abscabs 15126  π‘‚(1)co1 15375  β‰€π‘‚(1)clo1 15376  Ξ£csu 15577   βˆ₯ cdvds 16143  logclog 25926  Ξ›cvma 26457  Οˆcchp 26458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136  ax-addf 11137  ax-mulf 11138
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-fi 9354  df-sup 9385  df-inf 9386  df-oi 9453  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-xnn0 12493  df-z 12507  df-dec 12626  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-ioo 13275  df-ioc 13276  df-ico 13277  df-icc 13278  df-fz 13432  df-fzo 13575  df-fl 13704  df-mod 13782  df-seq 13914  df-exp 13975  df-fac 14181  df-bc 14210  df-hash 14238  df-shft 14959  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-limsup 15360  df-clim 15377  df-rlim 15378  df-o1 15379  df-lo1 15380  df-sum 15578  df-ef 15957  df-e 15958  df-sin 15959  df-cos 15960  df-tan 15961  df-pi 15962  df-dvds 16144  df-gcd 16382  df-prm 16555  df-pc 16716  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-ds 17162  df-unif 17163  df-hom 17164  df-cco 17165  df-rest 17311  df-topn 17312  df-0g 17330  df-gsum 17331  df-topgen 17332  df-pt 17333  df-prds 17336  df-xrs 17391  df-qtop 17396  df-imas 17397  df-xps 17399  df-mre 17473  df-mrc 17474  df-acs 17476  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-mulg 18880  df-cntz 19104  df-cmn 19571  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-fbas 20809  df-fg 20810  df-cnfld 20813  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-lp 22503  df-perf 22504  df-cn 22594  df-cnp 22595  df-haus 22682  df-cmp 22754  df-tx 22929  df-hmeo 23122  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307  df-xms 23689  df-ms 23690  df-tms 23691  df-cncf 24257  df-limc 25246  df-dv 25247  df-ulm 25752  df-log 25928  df-cxp 25929  df-atan 26233  df-em 26358  df-cht 26462  df-vma 26463  df-chp 26464  df-ppi 26465  df-mu 26466
This theorem is referenced by:  pntrlog2bndlem4  26944
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