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Theorem mulog2sumlem1 26024
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 13334 . . . . . 6 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 mulog2sumlem1.2 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ+)
3 elfznn 12929 . . . . . . . . . 10 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
43nnrpd 12422 . . . . . . . . 9 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5 rpdivcl 12407 . . . . . . . . 9 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (𝐴 / 𝑚) ∈ ℝ+)
62, 4, 5syl2an 595 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑚) ∈ ℝ+)
76relogcld 25119 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
83adantl 482 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
97, 8nndivred 11683 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
101, 9fsumrecl 15083 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
112relogcld 25119 . . . . . . . 8 (𝜑 → (log‘𝐴) ∈ ℝ)
1211resqcld 13604 . . . . . . 7 (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
1312rehalfcld 11876 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14 emre 25497 . . . . . . . 8 γ ∈ ℝ
15 remulcl 10614 . . . . . . . 8 ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
1614, 11, 15sylancr 587 . . . . . . 7 (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17 rpsup 13227 . . . . . . . . 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 26023 . . . . . . . . . . . 12 (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
2120simp1i 1133 . . . . . . . . . . 11 𝐹:ℝ+⟶ℝ
2221a1i 11 . . . . . . . . . 10 (𝜑𝐹:ℝ+⟶ℝ)
2322feqmptd 6729 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)))
24 mulog2sumlem.1 . . . . . . . . 9 (𝜑𝐹𝑟 𝐿)
2523, 24eqbrtrrd 5086 . . . . . . . 8 (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)) ⇝𝑟 𝐿)
2621ffvelrni 6845 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝐹𝑥) ∈ ℝ)
2726adantl 482 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → (𝐹𝑥) ∈ ℝ)
2818, 25, 27rlimrecl 14930 . . . . . . 7 (𝜑𝐿 ∈ ℝ)
2916, 28resubcld 11060 . . . . . 6 (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
3013, 29readdcld 10662 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
3110, 30resubcld 11060 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
3231recnd 10661 . . 3 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
3332abscld 14789 . 2 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34 rerpdivcl 12412 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ 𝑚 ∈ ℝ+) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3511, 4, 34syl2an 595 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3635recnd 10661 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
371, 36fsumcl 15082 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
3811recnd 10661 . . . . . 6 (𝜑 → (log‘𝐴) ∈ ℂ)
39 readdcl 10612 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) + γ) ∈ ℝ)
4011, 14, 39sylancl 586 . . . . . . 7 (𝜑 → ((log‘𝐴) + γ) ∈ ℝ)
4140recnd 10661 . . . . . 6 (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
4238, 41mulcld 10653 . . . . 5 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
4337, 42subcld 10989 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
4443abscld 14789 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
458nnrpd 12422 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
4645relogcld 25119 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
4746, 8nndivred 11683 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
4847recnd 10661 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
491, 48fsumcl 15082 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
5013recnd 10661 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
5128recnd 10661 . . . . . 6 (𝜑𝐿 ∈ ℂ)
5250, 51addcld 10652 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
5349, 52subcld 10989 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
5453abscld 14789 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
5544, 54readdcld 10662 . 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 11703 . . 3 2 ∈ ℝ
5711, 2rerpdivcld 12455 . . 3 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58 remulcl 10614 . . 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 25089 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
612, 4, 60syl2an 595 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
6261oveq1d 7166 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
6338adantr 481 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
6446recnd 10661 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
6545rpcnne0d 12433 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66 divsubdir 11326 . . . . . . . . . 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 1365 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6862, 67eqtrd 2860 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6968sumeq2dv 15052 . . . . . . 7 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
701, 36, 48fsumsub 15135 . . . . . . 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 2860 . . . . . 6 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72 remulcl 10614 . . . . . . . . . . . . 13 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) · γ) ∈ ℝ)
7311, 14, 72sylancl 586 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℝ)
7413, 73readdcld 10662 . . . . . . . . . . 11 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
7574recnd 10661 . . . . . . . . . 10 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
7675, 50pncand 10990 . . . . . . . . 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 10647 . . . . . . . . . . . . 13 γ ∈ ℂ
7877a1i 11 . . . . . . . . . . . 12 (𝜑 → γ ∈ ℂ)
7938, 38, 78adddid 10657 . . . . . . . . . . 11 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8012recnd 10661 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81802halvesd 11875 . . . . . . . . . . . . 13 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
8238sqvald 13500 . . . . . . . . . . . . 13 (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
8381, 82eqtrd 2860 . . . . . . . . . . . 12 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
8483oveq1d 7166 . . . . . . . . . . 11 (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8573recnd 10661 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
8650, 50, 85add32d 10859 . . . . . . . . . . 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 2866 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
8887oveq1d 7166 . . . . . . . . 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 10615 . . . . . . . . . . 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 7167 . . . . . . . . 9 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
9276, 88, 913eqtr4rd 2871 . . . . . . . 8 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
9392oveq1d 7166 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
9490, 85eqeltrd 2917 . . . . . . . 8 (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
9550, 94, 51addsubassd 11009 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
9642, 50, 51subsub4d 11020 . . . . . . 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 2868 . . . . . 6 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
9871, 97oveq12d 7169 . . . . 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 11038 . . . . 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 2860 . . . 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 6670 . . 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 14811 . . 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 5084 . 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 25502 . . . . . . 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 11680 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
1071, 106fsumrecl 15083 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108107, 40resubcld 11060 . . . . . . . . 9 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109108recnd 10661 . . . . . . . 8 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110109abscld 14789 . . . . . . 7 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
1112rprecred 12435 . . . . . . 7 (𝜑 → (1 / 𝐴) ∈ ℝ)
112 0red 10636 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
113 1red 10634 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
114 0lt1 11154 . . . . . . . . 9 0 < 1
115114a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
116 loge 25083 . . . . . . . . 9 (log‘e) = 1
117 mulog2sumlem1.3 . . . . . . . . . 10 (𝜑 → e ≤ 𝐴)
118 epr 15553 . . . . . . . . . . 11 e ∈ ℝ+
119 logleb 25099 . . . . . . . . . . 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 233 . . . . . . . . 9 (𝜑 → (log‘e) ≤ (log‘𝐴))
122116, 121eqbrtrrid 5098 . . . . . . . 8 (𝜑 → 1 ≤ (log‘𝐴))
123112, 113, 11, 115, 122ltletrd 10792 . . . . . . 7 (𝜑 → 0 < (log‘𝐴))
124 lemul2 11485 . . . . . . 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 1368 . . . . . 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 233 . . . . 5 (𝜑 → ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴)))
12745rpcnd 12426 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
12845rpne0d 12429 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
12963, 127, 128divrecd 11411 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130129sumeq2dv 15052 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131106recnd 10661 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
1321, 38, 131fsummulc2 15131 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133130, 132eqtr4d 2863 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134133oveq1d 7166 . . . . . . . 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 15082 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
13638, 135, 41subdid 11088 . . . . . . . 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 2863 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
138137fveq2d 6670 . . . . . 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 10989 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
14038, 139absmuld 14807 . . . . . 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 10780 . . . . . . . 8 (𝜑 → 0 ≤ (log‘𝐴))
14211, 141absidd 14775 . . . . . . 7 (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143142oveq1d 7166 . . . . . 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 2864 . . . . 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 12426 . . . . . 6 (𝜑𝐴 ∈ ℂ)
1462rpne0d 12429 . . . . . 6 (𝜑𝐴 ≠ 0)
14738, 145, 146divrecd 11411 . . . . 5 (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148126, 144, 1473brtr4d 5094 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149 fveq2 6666 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150 id 22 . . . . . . . . . . . . . 14 (𝑖 = 𝑚𝑖 = 𝑚)
151149, 150oveq12d 7169 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152151cbvsumv 15045 . . . . . . . . . . . 12 Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153 fveq2 6666 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154153oveq2d 7167 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155154sumeq1d 15050 . . . . . . . . . . . 12 (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156152, 155syl5eq 2872 . . . . . . . . . . 11 (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157 fveq2 6666 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158157oveq1d 7166 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159158oveq1d 7166 . . . . . . . . . . 11 (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160156, 159oveq12d 7169 . . . . . . . . . 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 7184 . . . . . . . . . 10 𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162160, 19, 161fvmpt 6764 . . . . . . . . 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 7166 . . . . . . 7 (𝜑 → ((𝐹𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
16549, 50, 51subsub4d 11020 . . . . . . 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 2860 . . . . . 6 (𝜑 → ((𝐹𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167166fveq2d 6670 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
16820simp3i 1135 . . . . . 6 ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
16924, 2, 117, 168syl3anc 1365 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170167, 169eqbrtrrd 5086 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ≤ ((log‘𝐴) / 𝐴))
17144, 54, 57, 57, 148, 170le2addd 11251 . . 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 10661 . . . 4 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
1731722timesd 11872 . . 3 (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174171, 173breqtrrd 5090 . 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 10789 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2106  wne 3020   class class class wbr 5062  cmpt 5142  dom cdm 5553  wf 6347  cfv 6351  (class class class)co 7151  supcsup 8896  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534  +∞cpnf 10664  *cxr 10666   < clt 10667  cle 10668  cmin 10862   / cdiv 11289  cn 11630  2c2 11684  +crp 12382  ...cfz 12885  cfl 13153  cexp 13422  abscabs 14586  𝑟 crli 14835  Σcsu 15035  eceu 15408  logclog 25051  γcem 25483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12383  df-xneg 12500  df-xadd 12501  df-xmul 12502  df-ioo 12735  df-ioc 12736  df-ico 12737  df-icc 12738  df-fz 12886  df-fzo 13027  df-fl 13155  df-mod 13231  df-seq 13363  df-exp 13423  df-fac 13627  df-bc 13656  df-hash 13684  df-shft 14419  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-limsup 14821  df-clim 14838  df-rlim 14839  df-sum 15036  df-ef 15413  df-e 15414  df-sin 15415  df-cos 15416  df-tan 15417  df-pi 15418  df-dvds 15600  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-submnd 17947  df-mulg 18157  df-cntz 18379  df-cmn 18830  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-mopn 20457  df-fbas 20458  df-fg 20459  df-cnfld 20462  df-top 21418  df-topon 21435  df-topsp 21457  df-bases 21470  df-cld 21543  df-ntr 21544  df-cls 21545  df-nei 21622  df-lp 21660  df-perf 21661  df-cn 21751  df-cnp 21752  df-haus 21839  df-cmp 21911  df-tx 22086  df-hmeo 22279  df-fil 22370  df-fm 22462  df-flim 22463  df-flf 22464  df-xms 22845  df-ms 22846  df-tms 22847  df-cncf 23401  df-limc 24379  df-dv 24380  df-ulm 24880  df-log 25053  df-cxp 25054  df-atan 25358  df-em 25484
This theorem is referenced by:  mulog2sumlem2  26025
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