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Theorem mulog2sumlem1 25118
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 12709 . . . . . 6 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 mulog2sumlem1.2 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ+)
3 elfznn 12309 . . . . . . . . . 10 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
43nnrpd 11814 . . . . . . . . 9 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5 rpdivcl 11800 . . . . . . . . 9 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (𝐴 / 𝑚) ∈ ℝ+)
62, 4, 5syl2an 494 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑚) ∈ ℝ+)
76relogcld 24268 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
83adantl 482 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
97, 8nndivred 11014 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
101, 9fsumrecl 14393 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
112relogcld 24268 . . . . . . . 8 (𝜑 → (log‘𝐴) ∈ ℝ)
1211resqcld 12972 . . . . . . 7 (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
1312rehalfcld 11224 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14 emre 24627 . . . . . . . 8 γ ∈ ℝ
15 remulcl 9966 . . . . . . . 8 ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
1614, 11, 15sylancr 694 . . . . . . 7 (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17 rpsup 12602 . . . . . . . . 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 25117 . . . . . . . . . . . 12 (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
2120simp1i 1068 . . . . . . . . . . 11 𝐹:ℝ+⟶ℝ
2221a1i 11 . . . . . . . . . 10 (𝜑𝐹:ℝ+⟶ℝ)
2322feqmptd 6207 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)))
24 mulog2sumlem.1 . . . . . . . . 9 (𝜑𝐹𝑟 𝐿)
2523, 24eqbrtrrd 4642 . . . . . . . 8 (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)) ⇝𝑟 𝐿)
2621ffvelrni 6315 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝐹𝑥) ∈ ℝ)
2726adantl 482 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → (𝐹𝑥) ∈ ℝ)
2818, 25, 27rlimrecl 14240 . . . . . . 7 (𝜑𝐿 ∈ ℝ)
2916, 28resubcld 10403 . . . . . 6 (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
3013, 29readdcld 10014 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
3110, 30resubcld 10403 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
3231recnd 10013 . . 3 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
3332abscld 14104 . 2 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34 rerpdivcl 11805 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ 𝑚 ∈ ℝ+) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3511, 4, 34syl2an 494 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3635recnd 10013 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
371, 36fsumcl 14392 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
3811recnd 10013 . . . . . 6 (𝜑 → (log‘𝐴) ∈ ℂ)
39 readdcl 9964 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) + γ) ∈ ℝ)
4011, 14, 39sylancl 693 . . . . . . 7 (𝜑 → ((log‘𝐴) + γ) ∈ ℝ)
4140recnd 10013 . . . . . 6 (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
4238, 41mulcld 10005 . . . . 5 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
4337, 42subcld 10337 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
4443abscld 14104 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
458nnrpd 11814 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
4645relogcld 24268 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
4746, 8nndivred 11014 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
4847recnd 10013 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
491, 48fsumcl 14392 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
5013recnd 10013 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
5128recnd 10013 . . . . . 6 (𝜑𝐿 ∈ ℂ)
5250, 51addcld 10004 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
5349, 52subcld 10337 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
5453abscld 14104 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
5544, 54readdcld 10014 . 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 11035 . . 3 2 ∈ ℝ
5711, 2rerpdivcld 11847 . . 3 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58 remulcl 9966 . . 3 ((2 ∈ ℝ ∧ ((log‘𝐴) / 𝐴) ∈ ℝ) → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
5956, 57, 58sylancr 694 . 2 (𝜑 → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
60 relogdiv 24238 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
612, 4, 60syl2an 494 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
6261oveq1d 6620 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
6338adantr 481 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
6446recnd 10013 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
6545rpcnne0d 11825 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66 divsubdir 10666 . . . . . . . . . 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 1323 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6862, 67eqtrd 2660 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6968sumeq2dv 14362 . . . . . . 7 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
701, 36, 48fsumsub 14443 . . . . . . 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 2660 . . . . . 6 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72 remulcl 9966 . . . . . . . . . . . . 13 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) · γ) ∈ ℝ)
7311, 14, 72sylancl 693 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℝ)
7413, 73readdcld 10014 . . . . . . . . . . 11 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
7574recnd 10013 . . . . . . . . . 10 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
7675, 50pncand 10338 . . . . . . . . 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 9997 . . . . . . . . . . . . 13 γ ∈ ℂ
7877a1i 11 . . . . . . . . . . . 12 (𝜑 → γ ∈ ℂ)
7938, 38, 78adddid 10009 . . . . . . . . . . 11 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8012recnd 10013 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81802halvesd 11223 . . . . . . . . . . . . 13 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
8238sqvald 12942 . . . . . . . . . . . . 13 (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
8381, 82eqtrd 2660 . . . . . . . . . . . 12 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
8483oveq1d 6620 . . . . . . . . . . 11 (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8573recnd 10013 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
8650, 50, 85add32d 10208 . . . . . . . . . . 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 2666 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
8887oveq1d 6620 . . . . . . . . 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 9967 . . . . . . . . . . 11 ((γ ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
9077, 38, 89sylancr 694 . . . . . . . . . 10 (𝜑 → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
9190oveq2d 6621 . . . . . . . . 9 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
9276, 88, 913eqtr4rd 2671 . . . . . . . 8 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
9392oveq1d 6620 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
9490, 85eqeltrd 2704 . . . . . . . 8 (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
9550, 94, 51addsubassd 10357 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
9642, 50, 51subsub4d 10368 . . . . . . 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 2668 . . . . . 6 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
9871, 97oveq12d 6623 . . . . 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 10386 . . . . 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 2660 . . . 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 6154 . . 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 14126 . . 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 4640 . 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 24632 . . . . . . 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 11011 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
1071, 106fsumrecl 14393 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108107, 40resubcld 10403 . . . . . . . . 9 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109108recnd 10013 . . . . . . . 8 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110109abscld 14104 . . . . . . 7 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
1112rprecred 11827 . . . . . . 7 (𝜑 → (1 / 𝐴) ∈ ℝ)
112 0red 9986 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
113 1red 10000 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
114 0lt1 10495 . . . . . . . . 9 0 < 1
115114a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
116 loge 24232 . . . . . . . . 9 (log‘e) = 1
117 mulog2sumlem1.3 . . . . . . . . . 10 (𝜑 → e ≤ 𝐴)
118 epr 14856 . . . . . . . . . . 11 e ∈ ℝ+
119 logleb 24248 . . . . . . . . . . 11 ((e ∈ ℝ+𝐴 ∈ ℝ+) → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
120118, 2, 119sylancr 694 . . . . . . . . . 10 (𝜑 → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
121117, 120mpbid 222 . . . . . . . . 9 (𝜑 → (log‘e) ≤ (log‘𝐴))
122116, 121syl5eqbrr 4654 . . . . . . . 8 (𝜑 → 1 ≤ (log‘𝐴))
123112, 113, 11, 115, 122ltletrd 10142 . . . . . . 7 (𝜑 → 0 < (log‘𝐴))
124 lemul2 10821 . . . . . . 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 1327 . . . . . 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 222 . . . . 5 (𝜑 → ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴)))
12745rpcnd 11818 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
12845rpne0d 11821 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
12963, 127, 128divrecd 10749 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130129sumeq2dv 14362 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131106recnd 10013 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
1321, 38, 131fsummulc2 14439 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133130, 132eqtr4d 2663 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134133oveq1d 6620 . . . . . . . 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 14392 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
13638, 135, 41subdid 10431 . . . . . . . 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 2663 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
138137fveq2d 6154 . . . . . 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 10337 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
14038, 139absmuld 14122 . . . . . 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 10130 . . . . . . . 8 (𝜑 → 0 ≤ (log‘𝐴))
14211, 141absidd 14090 . . . . . . 7 (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143142oveq1d 6620 . . . . . 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 2664 . . . . 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 11818 . . . . . 6 (𝜑𝐴 ∈ ℂ)
1462rpne0d 11821 . . . . . 6 (𝜑𝐴 ≠ 0)
14738, 145, 146divrecd 10749 . . . . 5 (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148126, 144, 1473brtr4d 4650 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149 fveq2 6150 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150 id 22 . . . . . . . . . . . . . 14 (𝑖 = 𝑚𝑖 = 𝑚)
151149, 150oveq12d 6623 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152151cbvsumv 14355 . . . . . . . . . . . 12 Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153 fveq2 6150 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154153oveq2d 6621 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155154sumeq1d 14360 . . . . . . . . . . . 12 (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156152, 155syl5eq 2672 . . . . . . . . . . 11 (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157 fveq2 6150 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158157oveq1d 6620 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159158oveq1d 6620 . . . . . . . . . . 11 (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160156, 159oveq12d 6623 . . . . . . . . . 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 6633 . . . . . . . . . 10 𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162160, 19, 161fvmpt 6240 . . . . . . . . 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 6620 . . . . . . 7 (𝜑 → ((𝐹𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
16549, 50, 51subsub4d 10368 . . . . . . 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 2660 . . . . . 6 (𝜑 → ((𝐹𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167166fveq2d 6154 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
16820simp3i 1070 . . . . . 6 ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
16924, 2, 117, 168syl3anc 1323 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170167, 169eqbrtrrd 4642 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ≤ ((log‘𝐴) / 𝐴))
17144, 54, 57, 57, 148, 170le2addd 10591 . . 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 10013 . . . 4 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
1731722timesd 11220 . . 3 (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174171, 173breqtrrd 4646 . 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 10139 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796   class class class wbr 4618  cmpt 4678  dom cdm 5079  wf 5846  cfv 5850  (class class class)co 6605  supcsup 8291  cc 9879  cr 9880  0cc0 9881  1c1 9882   + caddc 9884   · cmul 9886  +∞cpnf 10016  *cxr 10018   < clt 10019  cle 10020  cmin 10211   / cdiv 10629  cn 10965  2c2 11015  +crp 11776  ...cfz 12265  cfl 12528  cexp 12797  abscabs 13903  𝑟 crli 14145  Σcsu 14345  eceu 14713  logclog 24200  γcem 24613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959  ax-addf 9960  ax-mulf 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-fi 8262  df-sup 8293  df-inf 8294  df-oi 8360  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12118  df-ioc 12119  df-ico 12120  df-icc 12121  df-fz 12266  df-fzo 12404  df-fl 12530  df-mod 12606  df-seq 12739  df-exp 12798  df-fac 12998  df-bc 13027  df-hash 13055  df-shft 13736  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-limsup 14131  df-clim 14148  df-rlim 14149  df-sum 14346  df-ef 14718  df-e 14719  df-sin 14720  df-cos 14721  df-pi 14723  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-starv 15872  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-unif 15881  df-hom 15882  df-cco 15883  df-rest 15999  df-topn 16000  df-0g 16018  df-gsum 16019  df-topgen 16020  df-pt 16021  df-prds 16024  df-xrs 16078  df-qtop 16083  df-imas 16084  df-xps 16086  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-submnd 17252  df-mulg 17457  df-cntz 17666  df-cmn 18111  df-psmet 19652  df-xmet 19653  df-met 19654  df-bl 19655  df-mopn 19656  df-fbas 19657  df-fg 19658  df-cnfld 19661  df-top 20616  df-bases 20617  df-topon 20618  df-topsp 20619  df-cld 20728  df-ntr 20729  df-cls 20730  df-nei 20807  df-lp 20845  df-perf 20846  df-cn 20936  df-cnp 20937  df-haus 21024  df-cmp 21095  df-tx 21270  df-hmeo 21463  df-fil 21555  df-fm 21647  df-flim 21648  df-flf 21649  df-xms 22030  df-ms 22031  df-tms 22032  df-cncf 22584  df-limc 23531  df-dv 23532  df-log 24202  df-cxp 24203  df-em 24614
This theorem is referenced by:  mulog2sumlem2  25119
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