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Theorem mulog2sumlem2 26883
Description: Lemma for mulog2sum 26885. (Contributed by Mario Carneiro, 19-May-2016.)
Hypotheses
Ref Expression
logdivsum.1 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))
mulog2sumlem.1 (𝜑𝐹𝑟 𝐿)
mulog2sumlem2.t 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿))
mulog2sumlem2.r 𝑅 = (((1 / 2) + (γ + (abs‘𝐿))) + Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚))
Assertion
Ref Expression
mulog2sumlem2 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥))) ∈ 𝑂(1))
Distinct variable groups:   𝑖,𝑚,𝑛,𝑥,𝑦   𝑥,𝐹   𝑛,𝐿,𝑥   𝜑,𝑚,𝑛,𝑥   𝑅,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑖)   𝑅(𝑦,𝑖,𝑚)   𝑇(𝑥,𝑦,𝑖,𝑚,𝑛)   𝐹(𝑦,𝑖,𝑚,𝑛)   𝐿(𝑦,𝑖,𝑚)

Proof of Theorem mulog2sumlem2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 1red 11156 . 2 (𝜑 → 1 ∈ ℝ)
2 2re 12227 . . . 4 2 ∈ ℝ
3 fzfid 13878 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
4 simpr 485 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
5 elfznn 13470 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
65nnrpd 12955 . . . . . . . 8 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
7 rpdivcl 12940 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
84, 6, 7syl2an 596 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
98relogcld 25978 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
10 simplr 767 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
119, 10rerpdivcld 12988 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) / 𝑥) ∈ ℝ)
123, 11fsumrecl 15619 . . . 4 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥) ∈ ℝ)
13 remulcl 11136 . . . 4 ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ∈ ℝ)
142, 12, 13sylancr 587 . . 3 ((𝜑𝑥 ∈ ℝ+) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ∈ ℝ)
15 mulog2sumlem2.r . . . . . 6 𝑅 = (((1 / 2) + (γ + (abs‘𝐿))) + Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚))
16 halfre 12367 . . . . . . . 8 (1 / 2) ∈ ℝ
17 emre 26355 . . . . . . . . 9 γ ∈ ℝ
18 mulog2sumlem.1 . . . . . . . . . . 11 (𝜑𝐹𝑟 𝐿)
19 rlimcl 15385 . . . . . . . . . . 11 (𝐹𝑟 𝐿𝐿 ∈ ℂ)
2018, 19syl 17 . . . . . . . . . 10 (𝜑𝐿 ∈ ℂ)
2120abscld 15321 . . . . . . . . 9 (𝜑 → (abs‘𝐿) ∈ ℝ)
22 readdcl 11134 . . . . . . . . 9 ((γ ∈ ℝ ∧ (abs‘𝐿) ∈ ℝ) → (γ + (abs‘𝐿)) ∈ ℝ)
2317, 21, 22sylancr 587 . . . . . . . 8 (𝜑 → (γ + (abs‘𝐿)) ∈ ℝ)
24 readdcl 11134 . . . . . . . 8 (((1 / 2) ∈ ℝ ∧ (γ + (abs‘𝐿)) ∈ ℝ) → ((1 / 2) + (γ + (abs‘𝐿))) ∈ ℝ)
2516, 23, 24sylancr 587 . . . . . . 7 (𝜑 → ((1 / 2) + (γ + (abs‘𝐿))) ∈ ℝ)
26 fzfid 13878 . . . . . . . 8 (𝜑 → (1...2) ∈ Fin)
27 epr 16090 . . . . . . . . . . 11 e ∈ ℝ+
28 elfznn 13470 . . . . . . . . . . . . 13 (𝑚 ∈ (1...2) → 𝑚 ∈ ℕ)
2928adantl 482 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...2)) → 𝑚 ∈ ℕ)
3029nnrpd 12955 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...2)) → 𝑚 ∈ ℝ+)
31 rpdivcl 12940 . . . . . . . . . . 11 ((e ∈ ℝ+𝑚 ∈ ℝ+) → (e / 𝑚) ∈ ℝ+)
3227, 30, 31sylancr 587 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...2)) → (e / 𝑚) ∈ ℝ+)
3332relogcld 25978 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...2)) → (log‘(e / 𝑚)) ∈ ℝ)
3433, 29nndivred 12207 . . . . . . . 8 ((𝜑𝑚 ∈ (1...2)) → ((log‘(e / 𝑚)) / 𝑚) ∈ ℝ)
3526, 34fsumrecl 15619 . . . . . . 7 (𝜑 → Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚) ∈ ℝ)
3625, 35readdcld 11184 . . . . . 6 (𝜑 → (((1 / 2) + (γ + (abs‘𝐿))) + Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚)) ∈ ℝ)
3715, 36eqeltrid 2842 . . . . 5 (𝜑𝑅 ∈ ℝ)
38 remulcl 11136 . . . . 5 ((𝑅 ∈ ℝ ∧ 2 ∈ ℝ) → (𝑅 · 2) ∈ ℝ)
3937, 2, 38sylancl 586 . . . 4 (𝜑 → (𝑅 · 2) ∈ ℝ)
4039adantr 481 . . 3 ((𝜑𝑥 ∈ ℝ+) → (𝑅 · 2) ∈ ℝ)
412a1i 11 . . . 4 ((𝜑𝑥 ∈ ℝ+) → 2 ∈ ℝ)
42 rpssre 12922 . . . . 5 + ⊆ ℝ
43 2cnd 12231 . . . . 5 (𝜑 → 2 ∈ ℂ)
44 o1const 15502 . . . . 5 ((ℝ+ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ 2) ∈ 𝑂(1))
4542, 43, 44sylancr 587 . . . 4 (𝜑 → (𝑥 ∈ ℝ+ ↦ 2) ∈ 𝑂(1))
46 logfacrlim2 26574 . . . . 5 (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ⇝𝑟 1
47 rlimo1 15499 . . . . 5 ((𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ⇝𝑟 1 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ∈ 𝑂(1))
4846, 47mp1i 13 . . . 4 (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ∈ 𝑂(1))
4941, 12, 45, 48o1mul2 15507 . . 3 (𝜑 → (𝑥 ∈ ℝ+ ↦ (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥))) ∈ 𝑂(1))
5039recnd 11183 . . . 4 (𝜑 → (𝑅 · 2) ∈ ℂ)
51 o1const 15502 . . . 4 ((ℝ+ ⊆ ℝ ∧ (𝑅 · 2) ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ (𝑅 · 2)) ∈ 𝑂(1))
5242, 50, 51sylancr 587 . . 3 (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝑅 · 2)) ∈ 𝑂(1))
5314, 40, 49, 52o1add2 15506 . 2 (𝜑 → (𝑥 ∈ ℝ+ ↦ ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2))) ∈ 𝑂(1))
5414, 40readdcld 11184 . 2 ((𝜑𝑥 ∈ ℝ+) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)) ∈ ℝ)
555adantl 482 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
56 mucl 26490 . . . . . . . . 9 (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
5755, 56syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
5857zred 12607 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
5958, 55nndivred 12207 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
6059recnd 11183 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
61 mulog2sumlem2.t . . . . . 6 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿))
629recnd 11183 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
6362sqcld 14049 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
6463halfcld 12398 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) / 2) ∈ ℂ)
65 remulcl 11136 . . . . . . . . . 10 ((γ ∈ ℝ ∧ (log‘(𝑥 / 𝑛)) ∈ ℝ) → (γ · (log‘(𝑥 / 𝑛))) ∈ ℝ)
6617, 9, 65sylancr 587 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (γ · (log‘(𝑥 / 𝑛))) ∈ ℝ)
6766recnd 11183 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (γ · (log‘(𝑥 / 𝑛))) ∈ ℂ)
6820ad2antrr 724 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐿 ∈ ℂ)
6967, 68subcld 11512 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((γ · (log‘(𝑥 / 𝑛))) − 𝐿) ∈ ℂ)
7064, 69addcld 11174 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿)) ∈ ℂ)
7161, 70eqeltrid 2842 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
7260, 71mulcld 11175 . . . 4 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · 𝑇) ∈ ℂ)
733, 72fsumcl 15618 . . 3 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) ∈ ℂ)
74 relogcl 25931 . . . . 5 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
7574adantl 482 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
7675recnd 11183 . . 3 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
7773, 76subcld 11512 . 2 ((𝜑𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥)) ∈ ℂ)
7877abscld 15321 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥))) ∈ ℝ)
7978adantrr 715 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥))) ∈ ℝ)
8054adantrr 715 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)) ∈ ℝ)
8154recnd 11183 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)) ∈ ℂ)
8281abscld 15321 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2))) ∈ ℝ)
8382adantrr 715 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2))) ∈ ℝ)
8457zcnd 12608 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
85 fzfid 13878 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
86 elfznn 13470 . . . . . . . . . . . . . 14 (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
87 nnrp 12926 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+)
88 rpdivcl 12940 . . . . . . . . . . . . . . . . . 18 (((𝑥 / 𝑛) ∈ ℝ+𝑚 ∈ ℝ+) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ+)
898, 87, 88syl2an 596 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ+)
9089relogcld 25978 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (log‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
91 simpr 485 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
9290, 91nndivred 12207 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℝ)
9392recnd 11183 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℂ)
9486, 93sylan2 593 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℂ)
9585, 94fsumcl 15618 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℂ)
9671, 95subcld 11512 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) ∈ ℂ)
9755nncnd 12169 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
9855nnne0d 12203 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
9984, 96, 97, 98div23d 11968 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛) = (((μ‘𝑛) / 𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
10060, 71, 95subdid 11611 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) = ((((μ‘𝑛) / 𝑛) · 𝑇) − (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
10199, 100eqtrd 2776 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛) = ((((μ‘𝑛) / 𝑛) · 𝑇) − (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
102101sumeq2dv 15588 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · 𝑇) − (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
10360, 95mulcld 11175 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) ∈ ℂ)
1043, 72, 103fsumsub 15673 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · 𝑇) − (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
105102, 104eqtrd 2776 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
106105adantrr 715 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
10785, 60, 94fsummulc2 15669 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((μ‘𝑛) / 𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))
10884adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (μ‘𝑛) ∈ ℂ)
10997, 98jca 512 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
110109adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
111 div23 11832 . . . . . . . . . . . . . . . . 17 (((μ‘𝑛) ∈ ℂ ∧ ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((μ‘𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) / 𝑛) = (((μ‘𝑛) / 𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))
112 divass 11831 . . . . . . . . . . . . . . . . 17 (((μ‘𝑛) ∈ ℂ ∧ ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((μ‘𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) / 𝑛) = ((μ‘𝑛) · (((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) / 𝑛)))
113111, 112eqtr3d 2778 . . . . . . . . . . . . . . . 16 (((μ‘𝑛) ∈ ℂ ∧ ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((μ‘𝑛) / 𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = ((μ‘𝑛) · (((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) / 𝑛)))
114108, 93, 110, 113syl3anc 1371 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (((μ‘𝑛) / 𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = ((μ‘𝑛) · (((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) / 𝑛)))
11590recnd 11183 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (log‘((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
11691nnrpd 12955 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ+)
117 rpcnne0 12933 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℝ+ → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
118116, 117syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
119 divdiv1 11866 . . . . . . . . . . . . . . . . . 18 (((log‘((𝑥 / 𝑛) / 𝑚)) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0) ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) / 𝑛) = ((log‘((𝑥 / 𝑛) / 𝑚)) / (𝑚 · 𝑛)))
120115, 118, 110, 119syl3anc 1371 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) / 𝑛) = ((log‘((𝑥 / 𝑛) / 𝑚)) / (𝑚 · 𝑛)))
121 rpre 12923 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
122121adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
123122adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
124123recnd 11183 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
125124adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → 𝑥 ∈ ℂ)
126 divdiv1 11866 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → ((𝑥 / 𝑛) / 𝑚) = (𝑥 / (𝑛 · 𝑚)))
127125, 110, 118, 126syl3anc 1371 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → ((𝑥 / 𝑛) / 𝑚) = (𝑥 / (𝑛 · 𝑚)))
128127fveq2d 6846 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (log‘((𝑥 / 𝑛) / 𝑚)) = (log‘(𝑥 / (𝑛 · 𝑚))))
12991nncnd 12169 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
13097adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℂ)
131129, 130mulcomd 11176 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (𝑚 · 𝑛) = (𝑛 · 𝑚))
132128, 131oveq12d 7375 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → ((log‘((𝑥 / 𝑛) / 𝑚)) / (𝑚 · 𝑛)) = ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚)))
133120, 132eqtrd 2776 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) / 𝑛) = ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚)))
134133oveq2d 7373 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → ((μ‘𝑛) · (((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) / 𝑛)) = ((μ‘𝑛) · ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚))))
135114, 134eqtrd 2776 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (((μ‘𝑛) / 𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = ((μ‘𝑛) · ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚))))
13686, 135sylan2 593 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((μ‘𝑛) / 𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = ((μ‘𝑛) · ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚))))
137136sumeq2dv 15588 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((μ‘𝑛) / 𝑛) · ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚))))
138107, 137eqtrd 2776 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚))))
139138sumeq2dv 15588 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚))))
140 oveq2 7365 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 · 𝑚) → (𝑥 / 𝑘) = (𝑥 / (𝑛 · 𝑚)))
141140fveq2d 6846 . . . . . . . . . . . . 13 (𝑘 = (𝑛 · 𝑚) → (log‘(𝑥 / 𝑘)) = (log‘(𝑥 / (𝑛 · 𝑚))))
142 id 22 . . . . . . . . . . . . 13 (𝑘 = (𝑛 · 𝑚) → 𝑘 = (𝑛 · 𝑚))
143141, 142oveq12d 7375 . . . . . . . . . . . 12 (𝑘 = (𝑛 · 𝑚) → ((log‘(𝑥 / 𝑘)) / 𝑘) = ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚)))
144143oveq2d 7373 . . . . . . . . . . 11 (𝑘 = (𝑛 · 𝑚) → ((μ‘𝑛) · ((log‘(𝑥 / 𝑘)) / 𝑘)) = ((μ‘𝑛) · ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚))))
1454rpred 12957 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
146 ssrab2 4037 . . . . . . . . . . . . . . . 16 {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ ℕ
147 simprr 771 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})
148146, 147sselid 3942 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑛 ∈ ℕ)
149148, 56syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (μ‘𝑛) ∈ ℤ)
150149zred 12607 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (μ‘𝑛) ∈ ℝ)
151 elfznn 13470 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
152151adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → 𝑘 ∈ ℕ)
153152nnrpd 12955 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → 𝑘 ∈ ℝ+)
154 rpdivcl 12940 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑘 ∈ ℝ+) → (𝑥 / 𝑘) ∈ ℝ+)
1554, 153, 154syl2an 596 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (𝑥 / 𝑘) ∈ ℝ+)
156155relogcld 25978 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (log‘(𝑥 / 𝑘)) ∈ ℝ)
157151ad2antrl 726 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑘 ∈ ℕ)
158156, 157nndivred 12207 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → ((log‘(𝑥 / 𝑘)) / 𝑘) ∈ ℝ)
159150, 158remulcld 11185 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → ((μ‘𝑛) · ((log‘(𝑥 / 𝑘)) / 𝑘)) ∈ ℝ)
160159recnd 11183 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → ((μ‘𝑛) · ((log‘(𝑥 / 𝑘)) / 𝑘)) ∈ ℂ)
161144, 145, 160dvdsflsumcom 26537 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((μ‘𝑛) · ((log‘(𝑥 / 𝑘)) / 𝑘)) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘(𝑥 / (𝑛 · 𝑚))) / (𝑛 · 𝑚))))
162139, 161eqtr4d 2779 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((μ‘𝑛) · ((log‘(𝑥 / 𝑘)) / 𝑘)))
163162adantrr 715 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((μ‘𝑛) · ((log‘(𝑥 / 𝑘)) / 𝑘)))
164 oveq2 7365 . . . . . . . . . . 11 (𝑘 = 1 → (𝑥 / 𝑘) = (𝑥 / 1))
165164fveq2d 6846 . . . . . . . . . 10 (𝑘 = 1 → (log‘(𝑥 / 𝑘)) = (log‘(𝑥 / 1)))
166 id 22 . . . . . . . . . 10 (𝑘 = 1 → 𝑘 = 1)
167165, 166oveq12d 7375 . . . . . . . . 9 (𝑘 = 1 → ((log‘(𝑥 / 𝑘)) / 𝑘) = ((log‘(𝑥 / 1)) / 1))
168 fzfid 13878 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
169 fz1ssnn 13472 . . . . . . . . . 10 (1...(⌊‘𝑥)) ⊆ ℕ
170169a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ⊆ ℕ)
171122adantrr 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
172 simprr 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
173 flge1nn 13726 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
174171, 172, 173syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ)
175 nnuz 12806 . . . . . . . . . . 11 ℕ = (ℤ‘1)
176174, 175eleqtrdi 2848 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ (ℤ‘1))
177 eluzfz1 13448 . . . . . . . . . 10 ((⌊‘𝑥) ∈ (ℤ‘1) → 1 ∈ (1...(⌊‘𝑥)))
178176, 177syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ (1...(⌊‘𝑥)))
179151nnrpd 12955 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℝ+)
1804, 179, 154syl2an 596 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑘) ∈ ℝ+)
181180relogcld 25978 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑘)) ∈ ℝ)
182169a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ⊆ ℕ)
183182sselda 3944 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
184181, 183nndivred 12207 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑘)) / 𝑘) ∈ ℝ)
185184recnd 11183 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑘)) / 𝑘) ∈ ℂ)
186185adantlrr 719 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑘)) / 𝑘) ∈ ℂ)
187167, 168, 170, 178, 186musumsum 26541 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((μ‘𝑛) · ((log‘(𝑥 / 𝑘)) / 𝑘)) = ((log‘(𝑥 / 1)) / 1))
1884rpcnd 12959 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
189188div1d 11923 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℝ+) → (𝑥 / 1) = 𝑥)
190189fveq2d 6846 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ+) → (log‘(𝑥 / 1)) = (log‘𝑥))
191190oveq1d 7372 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ+) → ((log‘(𝑥 / 1)) / 1) = ((log‘𝑥) / 1))
19276div1d 11923 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ+) → ((log‘𝑥) / 1) = (log‘𝑥))
193191, 192eqtrd 2776 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ+) → ((log‘(𝑥 / 1)) / 1) = (log‘𝑥))
194193adantrr 715 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘(𝑥 / 1)) / 1) = (log‘𝑥))
195163, 187, 1943eqtrd 2780 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = (log‘𝑥))
196195oveq2d 7373 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥)))
197106, 196eqtrd 2776 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥)))
198197fveq2d 6846 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛)) = (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥))))
199 ere 15971 . . . . . . . . 9 e ∈ ℝ
200199a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → e ∈ ℝ)
201 1re 11155 . . . . . . . . 9 1 ∈ ℝ
202 1lt2 12324 . . . . . . . . . 10 1 < 2
203 egt2lt3 16088 . . . . . . . . . . 11 (2 < e ∧ e < 3)
204203simpli 484 . . . . . . . . . 10 2 < e
205201, 2, 199lttri 11281 . . . . . . . . . 10 ((1 < 2 ∧ 2 < e) → 1 < e)
206202, 204, 205mp2an 690 . . . . . . . . 9 1 < e
207201, 199, 206ltleii 11278 . . . . . . . 8 1 ≤ e
208200, 207jctir 521 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → (e ∈ ℝ ∧ 1 ≤ e))
20937adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → 𝑅 ∈ ℝ)
21016a1i 11 . . . . . . . . . . . 12 (𝜑 → (1 / 2) ∈ ℝ)
211 1rp 12919 . . . . . . . . . . . . . 14 1 ∈ ℝ+
212 rphalfcl 12942 . . . . . . . . . . . . . 14 (1 ∈ ℝ+ → (1 / 2) ∈ ℝ+)
213211, 212ax-mp 5 . . . . . . . . . . . . 13 (1 / 2) ∈ ℝ+
214 rpge0 12928 . . . . . . . . . . . . 13 ((1 / 2) ∈ ℝ+ → 0 ≤ (1 / 2))
215213, 214mp1i 13 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (1 / 2))
21617a1i 11 . . . . . . . . . . . . 13 (𝜑 → γ ∈ ℝ)
217 0re 11157 . . . . . . . . . . . . . . 15 0 ∈ ℝ
218 emgt0 26356 . . . . . . . . . . . . . . 15 0 < γ
219217, 17, 218ltleii 11278 . . . . . . . . . . . . . 14 0 ≤ γ
220219a1i 11 . . . . . . . . . . . . 13 (𝜑 → 0 ≤ γ)
22120absge0d 15329 . . . . . . . . . . . . 13 (𝜑 → 0 ≤ (abs‘𝐿))
222216, 21, 220, 221addge0d 11731 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (γ + (abs‘𝐿)))
223210, 23, 215, 222addge0d 11731 . . . . . . . . . . 11 (𝜑 → 0 ≤ ((1 / 2) + (γ + (abs‘𝐿))))
224 log1 25941 . . . . . . . . . . . . . 14 (log‘1) = 0
22529nncnd 12169 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (1...2)) → 𝑚 ∈ ℂ)
226225mulid2d 11173 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (1...2)) → (1 · 𝑚) = 𝑚)
22730rpred 12957 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (1...2)) → 𝑚 ∈ ℝ)
2282a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (1...2)) → 2 ∈ ℝ)
229199a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (1...2)) → e ∈ ℝ)
230 elfzle2 13445 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ (1...2) → 𝑚 ≤ 2)
231230adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (1...2)) → 𝑚 ≤ 2)
2322, 199, 204ltleii 11278 . . . . . . . . . . . . . . . . . . 19 2 ≤ e
233232a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (1...2)) → 2 ≤ e)
234227, 228, 229, 231, 233letrd 11312 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (1...2)) → 𝑚 ≤ e)
235226, 234eqbrtrd 5127 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ (1...2)) → (1 · 𝑚) ≤ e)
236 1red 11156 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (1...2)) → 1 ∈ ℝ)
237236, 229, 30lemuldivd 13006 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ (1...2)) → ((1 · 𝑚) ≤ e ↔ 1 ≤ (e / 𝑚)))
238235, 237mpbid 231 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ (1...2)) → 1 ≤ (e / 𝑚))
239 logleb 25958 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ+ ∧ (e / 𝑚) ∈ ℝ+) → (1 ≤ (e / 𝑚) ↔ (log‘1) ≤ (log‘(e / 𝑚))))
240211, 32, 239sylancr 587 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ (1...2)) → (1 ≤ (e / 𝑚) ↔ (log‘1) ≤ (log‘(e / 𝑚))))
241238, 240mpbid 231 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (1...2)) → (log‘1) ≤ (log‘(e / 𝑚)))
242224, 241eqbrtrrid 5141 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (1...2)) → 0 ≤ (log‘(e / 𝑚)))
24333, 30, 242divge0d 12997 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...2)) → 0 ≤ ((log‘(e / 𝑚)) / 𝑚))
24426, 34, 243fsumge0 15680 . . . . . . . . . . 11 (𝜑 → 0 ≤ Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚))
24525, 35, 223, 244addge0d 11731 . . . . . . . . . 10 (𝜑 → 0 ≤ (((1 / 2) + (γ + (abs‘𝐿))) + Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚)))
246245, 15breqtrrdi 5147 . . . . . . . . 9 (𝜑 → 0 ≤ 𝑅)
247246adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → 0 ≤ 𝑅)
248209, 247jca 512 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
24984, 96mulcld 11175 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ∈ ℂ)
250 remulcl 11136 . . . . . . . 8 ((2 ∈ ℝ ∧ ((log‘(𝑥 / 𝑛)) / 𝑥) ∈ ℝ) → (2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) ∈ ℝ)
2512, 11, 250sylancr 587 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) ∈ ℝ)
2522a1i 11 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℝ)
253 0le2 12255 . . . . . . . . 9 0 ≤ 2
254253a1i 11 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 2)
25597mulid2d 11173 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
256 fznnfl 13767 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
257122, 256syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ℝ+) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
258257simplbda 500 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛𝑥)
259255, 258eqbrtrd 5127 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
260 1red 11156 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
26155nnrpd 12955 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
262260, 123, 261lemuldivd 13006 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
263259, 262mpbid 231 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
264 logleb 25958 . . . . . . . . . . . 12 ((1 ∈ ℝ+ ∧ (𝑥 / 𝑛) ∈ ℝ+) → (1 ≤ (𝑥 / 𝑛) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑛))))
265211, 8, 264sylancr 587 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 ≤ (𝑥 / 𝑛) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑛))))
266263, 265mpbid 231 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘1) ≤ (log‘(𝑥 / 𝑛)))
267224, 266eqbrtrrid 5141 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘(𝑥 / 𝑛)))
268 rpregt0 12929 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
269268ad2antlr 725 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
270 divge0 12024 . . . . . . . . 9 ((((log‘(𝑥 / 𝑛)) ∈ ℝ ∧ 0 ≤ (log‘(𝑥 / 𝑛))) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((log‘(𝑥 / 𝑛)) / 𝑥))
2719, 267, 269, 270syl21anc 836 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘(𝑥 / 𝑛)) / 𝑥))
272252, 11, 254, 271mulge0d 11732 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (2 · ((log‘(𝑥 / 𝑛)) / 𝑥)))
273249abscld 15321 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ∈ ℝ)
274273adantr 481 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ∈ ℝ)
27596adantr 481 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) ∈ ℂ)
276275abscld 15321 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ∈ ℝ)
277261rpred 12957 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
278251, 277remulcld 11185 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) · 𝑛) ∈ ℝ)
279278adantr 481 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → ((2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) · 𝑛) ∈ ℝ)
28084, 96absmuld 15339 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) = ((abs‘(μ‘𝑛)) · (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))))
28184abscld 15321 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
28296abscld 15321 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ∈ ℝ)
28396absge0d 15329 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
284 mule1 26497 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (abs‘(μ‘𝑛)) ≤ 1)
28555, 284syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ≤ 1)
286281, 260, 282, 283, 285lemul1ad 12094 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) · (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ≤ (1 · (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))))
287282recnd 11183 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ∈ ℂ)
288287mulid2d 11173 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) = (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
289286, 288breqtrd 5131 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) · (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ≤ (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
290280, 289eqbrtrd 5127 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ≤ (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
291290adantr 481 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ≤ (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
292 logdivsum.1 . . . . . . . . . 10 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))
29318ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → 𝐹𝑟 𝐿)
2948adantr 481 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (𝑥 / 𝑛) ∈ ℝ+)
295 simpr 485 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → e ≤ (𝑥 / 𝑛))
296292, 293, 294, 295mulog2sumlem1 26882 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) − ((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿)))) ≤ (2 · ((log‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
29771, 95abssubd 15338 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) = (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) − 𝑇)))
298297adantr 481 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) = (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) − 𝑇)))
29961oveq2i 7368 . . . . . . . . . . 11 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) − 𝑇) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) − ((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿)))
300299fveq2i 6845 . . . . . . . . . 10 (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) − 𝑇)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) − ((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿))))
301298, 300eqtrdi 2792 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) = (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) − ((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿)))))
302 2cnd 12231 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
30311recnd 11183 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) / 𝑥) ∈ ℂ)
304302, 303, 97mulassd 11178 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) · 𝑛) = (2 · (((log‘(𝑥 / 𝑛)) / 𝑥) · 𝑛)))
305 rpcnne0 12933 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
306305ad2antlr 725 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
307 divdiv2 11867 . . . . . . . . . . . . . 14 (((log‘(𝑥 / 𝑛)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((log‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)) = (((log‘(𝑥 / 𝑛)) · 𝑛) / 𝑥))
30862, 306, 109, 307syl3anc 1371 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)) = (((log‘(𝑥 / 𝑛)) · 𝑛) / 𝑥))
309 div23 11832 . . . . . . . . . . . . . 14 (((log‘(𝑥 / 𝑛)) ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((log‘(𝑥 / 𝑛)) · 𝑛) / 𝑥) = (((log‘(𝑥 / 𝑛)) / 𝑥) · 𝑛))
31062, 97, 306, 309syl3anc 1371 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛)) · 𝑛) / 𝑥) = (((log‘(𝑥 / 𝑛)) / 𝑥) · 𝑛))
311308, 310eqtrd 2776 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)) = (((log‘(𝑥 / 𝑛)) / 𝑥) · 𝑛))
312311oveq2d 7373 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · ((log‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = (2 · (((log‘(𝑥 / 𝑛)) / 𝑥) · 𝑛)))
313304, 312eqtr4d 2779 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) · 𝑛) = (2 · ((log‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
314313adantr 481 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → ((2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) · 𝑛) = (2 · ((log‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
315296, 301, 3143brtr4d 5137 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ≤ ((2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) · 𝑛))
316274, 276, 279, 291, 315letrd 11312 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ e ≤ (𝑥 / 𝑛)) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ≤ ((2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) · 𝑛))
317273adantr 481 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ∈ ℝ)
318282adantr 481 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ∈ ℝ)
31937ad3antrrr 728 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → 𝑅 ∈ ℝ)
320290adantr 481 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ≤ (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
32171adantr 481 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → 𝑇 ∈ ℂ)
322321abscld 15321 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘𝑇) ∈ ℝ)
32395adantr 481 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℂ)
324323abscld 15321 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) ∈ ℝ)
325322, 324readdcld 11184 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((abs‘𝑇) + (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ∈ ℝ)
326321, 323abs2dif2d 15343 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ≤ ((abs‘𝑇) + (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))))
32725ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((1 / 2) + (γ + (abs‘𝐿))) ∈ ℝ)
32835ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚) ∈ ℝ)
32961fveq2i 6845 . . . . . . . . . . . 12 (abs‘𝑇) = (abs‘((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿)))
330329, 322eqeltrrid 2843 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿))) ∈ ℝ)
33164adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (((log‘(𝑥 / 𝑛))↑2) / 2) ∈ ℂ)
332331abscld 15321 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘(((log‘(𝑥 / 𝑛))↑2) / 2)) ∈ ℝ)
33369adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((γ · (log‘(𝑥 / 𝑛))) − 𝐿) ∈ ℂ)
334333abscld 15321 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((γ · (log‘(𝑥 / 𝑛))) − 𝐿)) ∈ ℝ)
335332, 334readdcld 11184 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((abs‘(((log‘(𝑥 / 𝑛))↑2) / 2)) + (abs‘((γ · (log‘(𝑥 / 𝑛))) − 𝐿))) ∈ ℝ)
336331, 333abstrid 15341 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿))) ≤ ((abs‘(((log‘(𝑥 / 𝑛))↑2) / 2)) + (abs‘((γ · (log‘(𝑥 / 𝑛))) − 𝐿))))
33716a1i 11 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (1 / 2) ∈ ℝ)
33823ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (γ + (abs‘𝐿)) ∈ ℝ)
3399resqcld 14030 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
340339rehalfcld 12400 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) / 2) ∈ ℝ)
3419sqge0d 14042 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘(𝑥 / 𝑛))↑2))
342 2pos 12256 . . . . . . . . . . . . . . . . . . . 20 0 < 2
3432, 342pm3.2i 471 . . . . . . . . . . . . . . . . . . 19 (2 ∈ ℝ ∧ 0 < 2)
344343a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 ∈ ℝ ∧ 0 < 2))
345 divge0 12024 . . . . . . . . . . . . . . . . . 18 (((((log‘(𝑥 / 𝑛))↑2) ∈ ℝ ∧ 0 ≤ ((log‘(𝑥 / 𝑛))↑2)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ (((log‘(𝑥 / 𝑛))↑2) / 2))
346339, 341, 344, 345syl21anc 836 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (((log‘(𝑥 / 𝑛))↑2) / 2))
347340, 346absidd 15307 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((log‘(𝑥 / 𝑛))↑2) / 2)) = (((log‘(𝑥 / 𝑛))↑2) / 2))
348347adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘(((log‘(𝑥 / 𝑛))↑2) / 2)) = (((log‘(𝑥 / 𝑛))↑2) / 2))
3498rpred 12957 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
350 ltle 11243 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 / 𝑛) ∈ ℝ ∧ e ∈ ℝ) → ((𝑥 / 𝑛) < e → (𝑥 / 𝑛) ≤ e))
351349, 199, 350sylancl 586 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) < e → (𝑥 / 𝑛) ≤ e))
352351imp 407 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (𝑥 / 𝑛) ≤ e)
3538adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (𝑥 / 𝑛) ∈ ℝ+)
354 logleb 25958 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 / 𝑛) ∈ ℝ+ ∧ e ∈ ℝ+) → ((𝑥 / 𝑛) ≤ e ↔ (log‘(𝑥 / 𝑛)) ≤ (log‘e)))
355353, 27, 354sylancl 586 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((𝑥 / 𝑛) ≤ e ↔ (log‘(𝑥 / 𝑛)) ≤ (log‘e)))
356352, 355mpbid 231 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (log‘(𝑥 / 𝑛)) ≤ (log‘e))
357 loge 25942 . . . . . . . . . . . . . . . . . . 19 (log‘e) = 1
358356, 357breqtrdi 5146 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (log‘(𝑥 / 𝑛)) ≤ 1)
359 0le1 11678 . . . . . . . . . . . . . . . . . . . . 21 0 ≤ 1
360359a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 1)
3619, 260, 267, 360le2sqd 14160 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) ≤ 1 ↔ ((log‘(𝑥 / 𝑛))↑2) ≤ (1↑2)))
362361adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((log‘(𝑥 / 𝑛)) ≤ 1 ↔ ((log‘(𝑥 / 𝑛))↑2) ≤ (1↑2)))
363358, 362mpbid 231 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((log‘(𝑥 / 𝑛))↑2) ≤ (1↑2))
364 sq1 14099 . . . . . . . . . . . . . . . . 17 (1↑2) = 1
365363, 364breqtrdi 5146 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((log‘(𝑥 / 𝑛))↑2) ≤ 1)
366339adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
367 1red 11156 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → 1 ∈ ℝ)
368343a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (2 ∈ ℝ ∧ 0 < 2))
369 lediv1 12020 . . . . . . . . . . . . . . . . 17 ((((log‘(𝑥 / 𝑛))↑2) ∈ ℝ ∧ 1 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((log‘(𝑥 / 𝑛))↑2) ≤ 1 ↔ (((log‘(𝑥 / 𝑛))↑2) / 2) ≤ (1 / 2)))
370366, 367, 368, 369syl3anc 1371 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (((log‘(𝑥 / 𝑛))↑2) ≤ 1 ↔ (((log‘(𝑥 / 𝑛))↑2) / 2) ≤ (1 / 2)))
371365, 370mpbid 231 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (((log‘(𝑥 / 𝑛))↑2) / 2) ≤ (1 / 2))
372348, 371eqbrtrd 5127 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘(((log‘(𝑥 / 𝑛))↑2) / 2)) ≤ (1 / 2))
37368abscld 15321 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝐿) ∈ ℝ)
37466, 373readdcld 11184 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((γ · (log‘(𝑥 / 𝑛))) + (abs‘𝐿)) ∈ ℝ)
375374adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((γ · (log‘(𝑥 / 𝑛))) + (abs‘𝐿)) ∈ ℝ)
37667adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (γ · (log‘(𝑥 / 𝑛))) ∈ ℂ)
37720ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → 𝐿 ∈ ℂ)
378376, 377abs2dif2d 15343 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((γ · (log‘(𝑥 / 𝑛))) − 𝐿)) ≤ ((abs‘(γ · (log‘(𝑥 / 𝑛)))) + (abs‘𝐿)))
37917a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → γ ∈ ℝ)
380219a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ γ)
381379, 9, 380, 267mulge0d 11732 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (γ · (log‘(𝑥 / 𝑛))))
38266, 381absidd 15307 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(γ · (log‘(𝑥 / 𝑛)))) = (γ · (log‘(𝑥 / 𝑛))))
383382adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘(γ · (log‘(𝑥 / 𝑛)))) = (γ · (log‘(𝑥 / 𝑛))))
384383oveq1d 7372 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((abs‘(γ · (log‘(𝑥 / 𝑛)))) + (abs‘𝐿)) = ((γ · (log‘(𝑥 / 𝑛))) + (abs‘𝐿)))
385378, 384breqtrd 5131 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((γ · (log‘(𝑥 / 𝑛))) − 𝐿)) ≤ ((γ · (log‘(𝑥 / 𝑛))) + (abs‘𝐿)))
38666adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (γ · (log‘(𝑥 / 𝑛))) ∈ ℝ)
38717a1i 11 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → γ ∈ ℝ)
388377abscld 15321 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘𝐿) ∈ ℝ)
3899adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
390387, 218jctir 521 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (γ ∈ ℝ ∧ 0 < γ))
391 lemul2 12008 . . . . . . . . . . . . . . . . . . 19 (((log‘(𝑥 / 𝑛)) ∈ ℝ ∧ 1 ∈ ℝ ∧ (γ ∈ ℝ ∧ 0 < γ)) → ((log‘(𝑥 / 𝑛)) ≤ 1 ↔ (γ · (log‘(𝑥 / 𝑛))) ≤ (γ · 1)))
392389, 367, 390, 391syl3anc 1371 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((log‘(𝑥 / 𝑛)) ≤ 1 ↔ (γ · (log‘(𝑥 / 𝑛))) ≤ (γ · 1)))
393358, 392mpbid 231 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (γ · (log‘(𝑥 / 𝑛))) ≤ (γ · 1))
39417recni 11169 . . . . . . . . . . . . . . . . . 18 γ ∈ ℂ
395394mulid1i 11159 . . . . . . . . . . . . . . . . 17 (γ · 1) = γ
396393, 395breqtrdi 5146 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (γ · (log‘(𝑥 / 𝑛))) ≤ γ)
397386, 387, 388, 396leadd1dd 11769 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((γ · (log‘(𝑥 / 𝑛))) + (abs‘𝐿)) ≤ (γ + (abs‘𝐿)))
398334, 375, 338, 385, 397letrd 11312 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((γ · (log‘(𝑥 / 𝑛))) − 𝐿)) ≤ (γ + (abs‘𝐿)))
399332, 334, 337, 338, 372, 398le2addd 11774 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((abs‘(((log‘(𝑥 / 𝑛))↑2) / 2)) + (abs‘((γ · (log‘(𝑥 / 𝑛))) − 𝐿))) ≤ ((1 / 2) + (γ + (abs‘𝐿))))
400330, 335, 327, 336, 399letrd 11312 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((((log‘(𝑥 / 𝑛))↑2) / 2) + ((γ · (log‘(𝑥 / 𝑛))) − 𝐿))) ≤ ((1 / 2) + (γ + (abs‘𝐿))))
401329, 400eqbrtrid 5140 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘𝑇) ≤ ((1 / 2) + (γ + (abs‘𝐿))))
40286, 92sylan2 593 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℝ)
40385, 402fsumrecl 15619 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℝ)
404403adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℝ)
40586, 90sylan2 593 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
40686, 129sylan2 593 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℂ)
407406mulid2d 11173 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 · 𝑚) = 𝑚)
408 fznnfl 13767 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 / 𝑛) ∈ ℝ → (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝑥 / 𝑛))))
409349, 408syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝑥 / 𝑛))))
410409simplbda 500 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ≤ (𝑥 / 𝑛))
411407, 410eqbrtrd 5127 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 · 𝑚) ≤ (𝑥 / 𝑛))
412 1red 11156 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 1 ∈ ℝ)
413349adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑥 / 𝑛) ∈ ℝ)
414116rpregt0d 12963 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ ℝ ∧ 0 < 𝑚))
41586, 414sylan2 593 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑚 ∈ ℝ ∧ 0 < 𝑚))
416 lemuldiv 12035 . . . . . . . . . . . . . . . . . . . 20 ((1 ∈ ℝ ∧ (𝑥 / 𝑛) ∈ ℝ ∧ (𝑚 ∈ ℝ ∧ 0 < 𝑚)) → ((1 · 𝑚) ≤ (𝑥 / 𝑛) ↔ 1 ≤ ((𝑥 / 𝑛) / 𝑚)))
417412, 413, 415, 416syl3anc 1371 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((1 · 𝑚) ≤ (𝑥 / 𝑛) ↔ 1 ≤ ((𝑥 / 𝑛) / 𝑚)))
418411, 417mpbid 231 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 1 ≤ ((𝑥 / 𝑛) / 𝑚))
41986, 89sylan2 593 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ+)
420 logleb 25958 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℝ+ ∧ ((𝑥 / 𝑛) / 𝑚) ∈ ℝ+) → (1 ≤ ((𝑥 / 𝑛) / 𝑚) ↔ (log‘1) ≤ (log‘((𝑥 / 𝑛) / 𝑚))))
421211, 419, 420sylancr 587 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 ≤ ((𝑥 / 𝑛) / 𝑚) ↔ (log‘1) ≤ (log‘((𝑥 / 𝑛) / 𝑚))))
422418, 421mpbid 231 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘1) ≤ (log‘((𝑥 / 𝑛) / 𝑚)))
423224, 422eqbrtrrid 5141 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 0 ≤ (log‘((𝑥 / 𝑛) / 𝑚)))
424 divge0 12024 . . . . . . . . . . . . . . . 16 ((((log‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ ∧ 0 ≤ (log‘((𝑥 / 𝑛) / 𝑚))) ∧ (𝑚 ∈ ℝ ∧ 0 < 𝑚)) → 0 ≤ ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))
425405, 423, 415, 424syl21anc 836 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 0 ≤ ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))
42685, 402, 425fsumge0 15680 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))
427426adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))
428404, 427absidd 15307 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))
429 fzfid 13878 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
430349flcld 13703 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℤ)
431430adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (⌊‘(𝑥 / 𝑛)) ∈ ℤ)
432 2z 12535 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℤ
433432a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → 2 ∈ ℤ)
434349adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (𝑥 / 𝑛) ∈ ℝ)
435199a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → e ∈ ℝ)
436 3re 12233 . . . . . . . . . . . . . . . . . . . . . . 23 3 ∈ ℝ
437436a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → 3 ∈ ℝ)
438 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (𝑥 / 𝑛) < e)
439203simpri 486 . . . . . . . . . . . . . . . . . . . . . . 23 e < 3
440439a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → e < 3)
441434, 435, 437, 438, 440lttrd 11316 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (𝑥 / 𝑛) < 3)
442 3z 12536 . . . . . . . . . . . . . . . . . . . . . 22 3 ∈ ℤ
443 fllt 13711 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 / 𝑛) ∈ ℝ ∧ 3 ∈ ℤ) → ((𝑥 / 𝑛) < 3 ↔ (⌊‘(𝑥 / 𝑛)) < 3))
444434, 442, 443sylancl 586 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((𝑥 / 𝑛) < 3 ↔ (⌊‘(𝑥 / 𝑛)) < 3))
445441, 444mpbid 231 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (⌊‘(𝑥 / 𝑛)) < 3)
446 df-3 12217 . . . . . . . . . . . . . . . . . . . 20 3 = (2 + 1)
447445, 446breqtrdi 5146 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (⌊‘(𝑥 / 𝑛)) < (2 + 1))
448 zleltp1 12554 . . . . . . . . . . . . . . . . . . . 20 (((⌊‘(𝑥 / 𝑛)) ∈ ℤ ∧ 2 ∈ ℤ) → ((⌊‘(𝑥 / 𝑛)) ≤ 2 ↔ (⌊‘(𝑥 / 𝑛)) < (2 + 1)))
449431, 432, 448sylancl 586 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((⌊‘(𝑥 / 𝑛)) ≤ 2 ↔ (⌊‘(𝑥 / 𝑛)) < (2 + 1)))
450447, 449mpbird 256 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (⌊‘(𝑥 / 𝑛)) ≤ 2)
451 eluz2 12769 . . . . . . . . . . . . . . . . . 18 (2 ∈ (ℤ‘(⌊‘(𝑥 / 𝑛))) ↔ ((⌊‘(𝑥 / 𝑛)) ∈ ℤ ∧ 2 ∈ ℤ ∧ (⌊‘(𝑥 / 𝑛)) ≤ 2))
452431, 433, 450, 451syl3anbrc 1343 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → 2 ∈ (ℤ‘(⌊‘(𝑥 / 𝑛))))
453 fzss2 13481 . . . . . . . . . . . . . . . . 17 (2 ∈ (ℤ‘(⌊‘(𝑥 / 𝑛))) → (1...(⌊‘(𝑥 / 𝑛))) ⊆ (1...2))
454452, 453syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (1...(⌊‘(𝑥 / 𝑛))) ⊆ (1...2))
455454sselda 3944 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ (1...2))
45634ad5ant15 757 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → ((log‘(e / 𝑚)) / 𝑚) ∈ ℝ)
457455, 456syldan 591 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘(e / 𝑚)) / 𝑚) ∈ ℝ)
458429, 457fsumrecl 15619 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘(e / 𝑚)) / 𝑚) ∈ ℝ)
45992adantlr 713 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ ℕ) → ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℝ)
46086, 459sylan2 593 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ∈ ℝ)
461352adantr 481 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → (𝑥 / 𝑛) ≤ e)
462434adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → (𝑥 / 𝑛) ∈ ℝ)
463199a1i 11 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → e ∈ ℝ)
46430rpregt0d 12963 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (1...2)) → (𝑚 ∈ ℝ ∧ 0 < 𝑚))
465464ad5ant15 757 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → (𝑚 ∈ ℝ ∧ 0 < 𝑚))
466 lediv1 12020 . . . . . . . . . . . . . . . . . . 19 (((𝑥 / 𝑛) ∈ ℝ ∧ e ∈ ℝ ∧ (𝑚 ∈ ℝ ∧ 0 < 𝑚)) → ((𝑥 / 𝑛) ≤ e ↔ ((𝑥 / 𝑛) / 𝑚) ≤ (e / 𝑚)))
467462, 463, 465, 466syl3anc 1371 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → ((𝑥 / 𝑛) ≤ e ↔ ((𝑥 / 𝑛) / 𝑚) ≤ (e / 𝑚)))
468461, 467mpbid 231 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → ((𝑥 / 𝑛) / 𝑚) ≤ (e / 𝑚))
46989adantlr 713 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ ℕ) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ+)
47028, 469sylan2 593 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ+)
47132ad5ant15 757 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → (e / 𝑚) ∈ ℝ+)
472470, 471logled 25982 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → (((𝑥 / 𝑛) / 𝑚) ≤ (e / 𝑚) ↔ (log‘((𝑥 / 𝑛) / 𝑚)) ≤ (log‘(e / 𝑚))))
473468, 472mpbid 231 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → (log‘((𝑥 / 𝑛) / 𝑚)) ≤ (log‘(e / 𝑚)))
47490adantlr 713 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ ℕ) → (log‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
47528, 474sylan2 593 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → (log‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
47633ad5ant15 757 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → (log‘(e / 𝑚)) ∈ ℝ)
477 lediv1 12020 . . . . . . . . . . . . . . . . 17 (((log‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ ∧ (log‘(e / 𝑚)) ∈ ℝ ∧ (𝑚 ∈ ℝ ∧ 0 < 𝑚)) → ((log‘((𝑥 / 𝑛) / 𝑚)) ≤ (log‘(e / 𝑚)) ↔ ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ≤ ((log‘(e / 𝑚)) / 𝑚)))
478475, 476, 465, 477syl3anc 1371 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → ((log‘((𝑥 / 𝑛) / 𝑚)) ≤ (log‘(e / 𝑚)) ↔ ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ≤ ((log‘(e / 𝑚)) / 𝑚)))
479473, 478mpbid 231 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ≤ ((log‘(e / 𝑚)) / 𝑚))
480455, 479syldan 591 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ≤ ((log‘(e / 𝑚)) / 𝑚))
481429, 460, 457, 480fsumle 15684 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘(e / 𝑚)) / 𝑚))
482 fzfid 13878 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (1...2) ∈ Fin)
483243ad5ant15 757 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) ∧ 𝑚 ∈ (1...2)) → 0 ≤ ((log‘(e / 𝑚)) / 𝑚))
484482, 456, 483, 454fsumless 15681 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘(e / 𝑚)) / 𝑚) ≤ Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚))
485404, 458, 328, 481, 484letrd 11312 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚) ≤ Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚))
486428, 485eqbrtrd 5127 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)) ≤ Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚))
487322, 324, 327, 328, 401, 486le2addd 11774 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((abs‘𝑇) + (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ≤ (((1 / 2) + (γ + (abs‘𝐿))) + Σ𝑚 ∈ (1...2)((log‘(e / 𝑚)) / 𝑚)))
488487, 15breqtrrdi 5147 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → ((abs‘𝑇) + (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ≤ 𝑅)
489318, 325, 319, 326, 488letrd 11312 . . . . . . . 8 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘(𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) ≤ 𝑅)
490317, 318, 319, 320, 489letrd 11312 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ (𝑥 / 𝑛) < e) → (abs‘((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚)))) ≤ 𝑅)
4914, 208, 248, 249, 251, 272, 316, 490fsumharmonic 26361 . . . . . 6 ((𝜑𝑥 ∈ ℝ+) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · ((log‘e) + 1))))
492 2cnd 12231 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → 2 ∈ ℂ)
4933, 492, 303fsummulc2 15669 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(2 · ((log‘(𝑥 / 𝑛)) / 𝑥)))
494 df-2 12216 . . . . . . . . . 10 2 = (1 + 1)
495357oveq1i 7367 . . . . . . . . . 10 ((log‘e) + 1) = (1 + 1)
496494, 495eqtr4i 2767 . . . . . . . . 9 2 = ((log‘e) + 1)
497496a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → 2 = ((log‘e) + 1))
498497oveq2d 7373 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → (𝑅 · 2) = (𝑅 · ((log‘e) + 1)))
499493, 498oveq12d 7375 . . . . . 6 ((𝜑𝑥 ∈ ℝ+) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(2 · ((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · ((log‘e) + 1))))
500491, 499breqtrrd 5133 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛)) ≤ ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)))
501500adantrr 715 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · (𝑇 − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘((𝑥 / 𝑛) / 𝑚)) / 𝑚))) / 𝑛)) ≤ ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)))
502198, 501eqbrtrrd 5129 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥))) ≤ ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)))
50354leabsd 15299 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)) ≤ (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2))))
504503adantrr 715 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2)) ≤ (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2))))
50579, 80, 83, 502, 504letrd 11312 . 2 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥))) ≤ (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) + (𝑅 · 2))))
5061, 53, 54, 77, 505o1le 15537 1 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · 𝑇) − (log‘𝑥))) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  {crab 3407  wss 3910   class class class wbr 5105  cmpt 5188  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  2c2 12208  3c3 12209  cz 12499  cuz 12763  +crp 12915  ...cfz 13424  cfl 13695  cexp 13967  abscabs 15119  𝑟 crli 15367  𝑂(1)co1 15368  Σcsu 15570  eceu 15945  cdvds 16136  logclog 25910  γcem 26341  μcmu 26444
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-o1 15372  df-lo1 15373  df-sum 15571  df-ef 15950  df-e 15951  df-sin 15952  df-cos 15953  df-tan 15954  df-pi 15955  df-dvds 16137  df-gcd 16375  df-prm 16548  df-pc 16709  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231  df-ulm 25736  df-log 25912  df-cxp 25913  df-atan 26217  df-em 26342  df-mu 26450
This theorem is referenced by:  mulog2sumlem3  26884
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