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Theorem mulog2sumlem1 27593
Description: Asymptotic formula for Σ𝑛𝑥, log(𝑥 / 𝑛) / 𝑛 = (1 / 2)log↑2(𝑥) + γ · log𝑥𝐿 + 𝑂(log𝑥 / 𝑥), with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
logdivsum.1 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))
mulog2sumlem.1 (𝜑𝐹𝑟 𝐿)
mulog2sumlem1.2 (𝜑𝐴 ∈ ℝ+)
mulog2sumlem1.3 (𝜑 → e ≤ 𝐴)
Assertion
Ref Expression
mulog2sumlem1 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ≤ (2 · ((log‘𝐴) / 𝐴)))
Distinct variable groups:   𝑖,𝑚,𝑦,𝐴   𝜑,𝑚
Allowed substitution hints:   𝜑(𝑦,𝑖)   𝐹(𝑦,𝑖,𝑚)   𝐿(𝑦,𝑖,𝑚)

Proof of Theorem mulog2sumlem1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fzfid 14011 . . . . . 6 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 mulog2sumlem1.2 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ+)
3 elfznn 13590 . . . . . . . . . 10 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
43nnrpd 13073 . . . . . . . . 9 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5 rpdivcl 13058 . . . . . . . . 9 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (𝐴 / 𝑚) ∈ ℝ+)
62, 4, 5syl2an 596 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑚) ∈ ℝ+)
76relogcld 26680 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
83adantl 481 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
97, 8nndivred 12318 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
101, 9fsumrecl 15767 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
112relogcld 26680 . . . . . . . 8 (𝜑 → (log‘𝐴) ∈ ℝ)
1211resqcld 14162 . . . . . . 7 (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
1312rehalfcld 12511 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14 emre 27064 . . . . . . . 8 γ ∈ ℝ
15 remulcl 11238 . . . . . . . 8 ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
1614, 11, 15sylancr 587 . . . . . . 7 (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17 rpsup 13903 . . . . . . . . 9 sup(ℝ+, ℝ*, < ) = +∞
1817a1i 11 . . . . . . . 8 (𝜑 → sup(ℝ+, ℝ*, < ) = +∞)
19 logdivsum.1 . . . . . . . . . . . . 13 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))
2019logdivsum 27592 . . . . . . . . . . . 12 (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
2120simp1i 1138 . . . . . . . . . . 11 𝐹:ℝ+⟶ℝ
2221a1i 11 . . . . . . . . . 10 (𝜑𝐹:ℝ+⟶ℝ)
2322feqmptd 6977 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)))
24 mulog2sumlem.1 . . . . . . . . 9 (𝜑𝐹𝑟 𝐿)
2523, 24eqbrtrrd 5172 . . . . . . . 8 (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)) ⇝𝑟 𝐿)
2621ffvelcdmi 7103 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝐹𝑥) ∈ ℝ)
2726adantl 481 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → (𝐹𝑥) ∈ ℝ)
2818, 25, 27rlimrecl 15613 . . . . . . 7 (𝜑𝐿 ∈ ℝ)
2916, 28resubcld 11689 . . . . . 6 (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
3013, 29readdcld 11288 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
3110, 30resubcld 11689 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
3231recnd 11287 . . 3 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
3332abscld 15472 . 2 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34 rerpdivcl 13063 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ 𝑚 ∈ ℝ+) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3511, 4, 34syl2an 596 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3635recnd 11287 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
371, 36fsumcl 15766 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
3811recnd 11287 . . . . . 6 (𝜑 → (log‘𝐴) ∈ ℂ)
39 readdcl 11236 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) + γ) ∈ ℝ)
4011, 14, 39sylancl 586 . . . . . . 7 (𝜑 → ((log‘𝐴) + γ) ∈ ℝ)
4140recnd 11287 . . . . . 6 (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
4238, 41mulcld 11279 . . . . 5 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
4337, 42subcld 11618 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
4443abscld 15472 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
458nnrpd 13073 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
4645relogcld 26680 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
4746, 8nndivred 12318 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
4847recnd 11287 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
491, 48fsumcl 15766 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
5013recnd 11287 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
5128recnd 11287 . . . . . 6 (𝜑𝐿 ∈ ℂ)
5250, 51addcld 11278 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
5349, 52subcld 11618 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
5453abscld 15472 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
5544, 54readdcld 11288 . 2 (𝜑 → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))) ∈ ℝ)
56 2re 12338 . . 3 2 ∈ ℝ
5711, 2rerpdivcld 13106 . . 3 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58 remulcl 11238 . . 3 ((2 ∈ ℝ ∧ ((log‘𝐴) / 𝐴) ∈ ℝ) → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
5956, 57, 58sylancr 587 . 2 (𝜑 → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
60 relogdiv 26650 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
612, 4, 60syl2an 596 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
6261oveq1d 7446 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
6338adantr 480 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
6446recnd 11287 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
6545rpcnne0d 13084 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66 divsubdir 11959 . . . . . . . . . 10 (((log‘𝐴) ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6763, 64, 65, 66syl3anc 1370 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6862, 67eqtrd 2775 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6968sumeq2dv 15735 . . . . . . 7 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
701, 36, 48fsumsub 15821 . . . . . . 7 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
7169, 70eqtrd 2775 . . . . . 6 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72 remulcl 11238 . . . . . . . . . . . . 13 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) · γ) ∈ ℝ)
7311, 14, 72sylancl 586 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℝ)
7413, 73readdcld 11288 . . . . . . . . . . 11 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
7574recnd 11287 . . . . . . . . . 10 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
7675, 50pncand 11619 . . . . . . . . 9 (𝜑 → ((((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)) − (((log‘𝐴)↑2) / 2)) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
7714recni 11273 . . . . . . . . . . . . 13 γ ∈ ℂ
7877a1i 11 . . . . . . . . . . . 12 (𝜑 → γ ∈ ℂ)
7938, 38, 78adddid 11283 . . . . . . . . . . 11 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8012recnd 11287 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81802halvesd 12510 . . . . . . . . . . . . 13 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
8238sqvald 14180 . . . . . . . . . . . . 13 (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
8381, 82eqtrd 2775 . . . . . . . . . . . 12 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
8483oveq1d 7446 . . . . . . . . . . 11 (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8573recnd 11287 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
8650, 50, 85add32d 11487 . . . . . . . . . . 11 (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
8779, 84, 863eqtr2d 2781 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
8887oveq1d 7446 . . . . . . . . 9 (𝜑 → (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) = ((((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)) − (((log‘𝐴)↑2) / 2)))
89 mulcom 11239 . . . . . . . . . . 11 ((γ ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
9077, 38, 89sylancr 587 . . . . . . . . . 10 (𝜑 → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
9190oveq2d 7447 . . . . . . . . 9 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
9276, 88, 913eqtr4rd 2786 . . . . . . . 8 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
9392oveq1d 7446 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
9490, 85eqeltrd 2839 . . . . . . . 8 (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
9550, 94, 51addsubassd 11638 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
9642, 50, 51subsub4d 11649 . . . . . . 7 (𝜑 → ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
9793, 95, 963eqtr3d 2783 . . . . . 6 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
9871, 97oveq12d 7449 . . . . 5 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)) − (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
9937, 49, 42, 52sub4d 11667 . . . . 5 (𝜑 → ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)) − (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿))) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) − (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
10098, 99eqtrd 2775 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) − (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
101100fveq2d 6911 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) = (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) − (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))))
10243, 53abs2dif2d 15494 . . 3 (𝜑 → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) − (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))) ≤ ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))))
103101, 102eqbrtrd 5170 . 2 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ≤ ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))))
104 harmonicbnd4 27069 . . . . . . 7 (𝐴 ∈ ℝ+ → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))
1052, 104syl 17 . . . . . 6 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))
1068nnrecred 12315 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
1071, 106fsumrecl 15767 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108107, 40resubcld 11689 . . . . . . . . 9 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109108recnd 11287 . . . . . . . 8 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110109abscld 15472 . . . . . . 7 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
1112rprecred 13086 . . . . . . 7 (𝜑 → (1 / 𝐴) ∈ ℝ)
112 0red 11262 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
113 1red 11260 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
114 0lt1 11783 . . . . . . . . 9 0 < 1
115114a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
116 loge 26643 . . . . . . . . 9 (log‘e) = 1
117 mulog2sumlem1.3 . . . . . . . . . 10 (𝜑 → e ≤ 𝐴)
118 epr 16241 . . . . . . . . . . 11 e ∈ ℝ+
119 logleb 26660 . . . . . . . . . . 11 ((e ∈ ℝ+𝐴 ∈ ℝ+) → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
120118, 2, 119sylancr 587 . . . . . . . . . 10 (𝜑 → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
121117, 120mpbid 232 . . . . . . . . 9 (𝜑 → (log‘e) ≤ (log‘𝐴))
122116, 121eqbrtrrid 5184 . . . . . . . 8 (𝜑 → 1 ≤ (log‘𝐴))
123112, 113, 11, 115, 122ltletrd 11419 . . . . . . 7 (𝜑 → 0 < (log‘𝐴))
124 lemul2 12118 . . . . . . 7 (((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ ∧ ((log‘𝐴) ∈ ℝ ∧ 0 < (log‘𝐴))) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴) ↔ ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴))))
125110, 111, 11, 123, 124syl112anc 1373 . . . . . 6 (𝜑 → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴) ↔ ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴))))
126105, 125mpbid 232 . . . . 5 (𝜑 → ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴)))
12745rpcnd 13077 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
12845rpne0d 13080 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
12963, 127, 128divrecd 12044 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130129sumeq2dv 15735 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131106recnd 11287 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
1321, 38, 131fsummulc2 15817 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133130, 132eqtr4d 2778 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134133oveq1d 7446 . . . . . . . 8 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = (((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) − ((log‘𝐴) · ((log‘𝐴) + γ))))
1351, 131fsumcl 15766 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
13638, 135, 41subdid 11717 . . . . . . . 8 (𝜑 → ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) = (((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) − ((log‘𝐴) · ((log‘𝐴) + γ))))
137134, 136eqtr4d 2778 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
138137fveq2d 6911 . . . . . 6 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) = (abs‘((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))))
139135, 41subcld 11618 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
14038, 139absmuld 15490 . . . . . 6 (𝜑 → (abs‘((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) = ((abs‘(log‘𝐴)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))))
141112, 11, 123ltled 11407 . . . . . . . 8 (𝜑 → 0 ≤ (log‘𝐴))
14211, 141absidd 15458 . . . . . . 7 (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143142oveq1d 7446 . . . . . 6 (𝜑 → ((abs‘(log‘𝐴)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) = ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))))
144138, 140, 1433eqtrd 2779 . . . . 5 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) = ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))))
1452rpcnd 13077 . . . . . 6 (𝜑𝐴 ∈ ℂ)
1462rpne0d 13080 . . . . . 6 (𝜑𝐴 ≠ 0)
14738, 145, 146divrecd 12044 . . . . 5 (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148126, 144, 1473brtr4d 5180 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149 fveq2 6907 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150 id 22 . . . . . . . . . . . . . 14 (𝑖 = 𝑚𝑖 = 𝑚)
151149, 150oveq12d 7449 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152151cbvsumv 15729 . . . . . . . . . . . 12 Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153 fveq2 6907 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154153oveq2d 7447 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155154sumeq1d 15733 . . . . . . . . . . . 12 (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156152, 155eqtrid 2787 . . . . . . . . . . 11 (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157 fveq2 6907 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158157oveq1d 7446 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159158oveq1d 7446 . . . . . . . . . . 11 (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160156, 159oveq12d 7449 . . . . . . . . . 10 (𝑦 = 𝐴 → (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)))
161 ovex 7464 . . . . . . . . . 10 𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162160, 19, 161fvmpt 7016 . . . . . . . . 9 (𝐴 ∈ ℝ+ → (𝐹𝐴) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)))
1632, 162syl 17 . . . . . . . 8 (𝜑 → (𝐹𝐴) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)))
164163oveq1d 7446 . . . . . . 7 (𝜑 → ((𝐹𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
16549, 50, 51subsub4d 11649 . . . . . . 7 (𝜑 → ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
166164, 165eqtrd 2775 . . . . . 6 (𝜑 → ((𝐹𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167166fveq2d 6911 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
16820simp3i 1140 . . . . . 6 ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
16924, 2, 117, 168syl3anc 1370 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170167, 169eqbrtrrd 5172 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ≤ ((log‘𝐴) / 𝐴))
17144, 54, 57, 57, 148, 170le2addd 11880 . . 3 (𝜑 → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))) ≤ (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
17257recnd 11287 . . . 4 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
1731722timesd 12507 . . 3 (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174171, 173breqtrrd 5176 . 2 (𝜑 → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))) ≤ (2 · ((log‘𝐴) / 𝐴)))
17533, 55, 59, 103, 174letrd 11416 1 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ≤ (2 · ((log‘𝐴) / 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  supcsup 9478  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290  *cxr 11292   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  2c2 12319  +crp 13032  ...cfz 13544  cfl 13827  cexp 14099  abscabs 15270  𝑟 crli 15518  Σcsu 15719  eceu 16095  logclog 26611  γcem 27050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ioc 13389  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15103  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-sum 15720  df-ef 16100  df-e 16101  df-sin 16102  df-cos 16103  df-tan 16104  df-pi 16105  df-dvds 16288  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-ulm 26435  df-log 26613  df-cxp 26614  df-atan 26925  df-em 27051
This theorem is referenced by:  mulog2sumlem2  27594
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