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Theorem mulog2sumlem1 26882
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 13878 . . . . . 6 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 mulog2sumlem1.2 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ+)
3 elfznn 13470 . . . . . . . . . 10 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
43nnrpd 12955 . . . . . . . . 9 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5 rpdivcl 12940 . . . . . . . . 9 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (𝐴 / 𝑚) ∈ ℝ+)
62, 4, 5syl2an 596 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑚) ∈ ℝ+)
76relogcld 25978 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
83adantl 482 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
97, 8nndivred 12207 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
101, 9fsumrecl 15619 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
112relogcld 25978 . . . . . . . 8 (𝜑 → (log‘𝐴) ∈ ℝ)
1211resqcld 14030 . . . . . . 7 (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
1312rehalfcld 12400 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14 emre 26355 . . . . . . . 8 γ ∈ ℝ
15 remulcl 11136 . . . . . . . 8 ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
1614, 11, 15sylancr 587 . . . . . . 7 (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17 rpsup 13771 . . . . . . . . 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 26881 . . . . . . . . . . . 12 (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
2120simp1i 1139 . . . . . . . . . . 11 𝐹:ℝ+⟶ℝ
2221a1i 11 . . . . . . . . . 10 (𝜑𝐹:ℝ+⟶ℝ)
2322feqmptd 6910 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)))
24 mulog2sumlem.1 . . . . . . . . 9 (𝜑𝐹𝑟 𝐿)
2523, 24eqbrtrrd 5129 . . . . . . . 8 (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)) ⇝𝑟 𝐿)
2621ffvelcdmi 7034 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝐹𝑥) ∈ ℝ)
2726adantl 482 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → (𝐹𝑥) ∈ ℝ)
2818, 25, 27rlimrecl 15462 . . . . . . 7 (𝜑𝐿 ∈ ℝ)
2916, 28resubcld 11583 . . . . . 6 (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
3013, 29readdcld 11184 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
3110, 30resubcld 11583 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
3231recnd 11183 . . 3 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
3332abscld 15321 . 2 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34 rerpdivcl 12945 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ 𝑚 ∈ ℝ+) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3511, 4, 34syl2an 596 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3635recnd 11183 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
371, 36fsumcl 15618 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
3811recnd 11183 . . . . . 6 (𝜑 → (log‘𝐴) ∈ ℂ)
39 readdcl 11134 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) + γ) ∈ ℝ)
4011, 14, 39sylancl 586 . . . . . . 7 (𝜑 → ((log‘𝐴) + γ) ∈ ℝ)
4140recnd 11183 . . . . . 6 (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
4238, 41mulcld 11175 . . . . 5 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
4337, 42subcld 11512 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
4443abscld 15321 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
458nnrpd 12955 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
4645relogcld 25978 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
4746, 8nndivred 12207 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
4847recnd 11183 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
491, 48fsumcl 15618 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
5013recnd 11183 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
5128recnd 11183 . . . . . 6 (𝜑𝐿 ∈ ℂ)
5250, 51addcld 11174 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
5349, 52subcld 11512 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
5453abscld 15321 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
5544, 54readdcld 11184 . 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 12227 . . 3 2 ∈ ℝ
5711, 2rerpdivcld 12988 . . 3 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58 remulcl 11136 . . 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 25948 . . . . . . . . . . 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 7372 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
6338adantr 481 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
6446recnd 11183 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
6545rpcnne0d 12966 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66 divsubdir 11849 . . . . . . . . . 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 1371 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6862, 67eqtrd 2776 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6968sumeq2dv 15588 . . . . . . 7 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
701, 36, 48fsumsub 15673 . . . . . . 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 2776 . . . . . 6 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72 remulcl 11136 . . . . . . . . . . . . 13 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) · γ) ∈ ℝ)
7311, 14, 72sylancl 586 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℝ)
7413, 73readdcld 11184 . . . . . . . . . . 11 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
7574recnd 11183 . . . . . . . . . 10 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
7675, 50pncand 11513 . . . . . . . . 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 11169 . . . . . . . . . . . . 13 γ ∈ ℂ
7877a1i 11 . . . . . . . . . . . 12 (𝜑 → γ ∈ ℂ)
7938, 38, 78adddid 11179 . . . . . . . . . . 11 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8012recnd 11183 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81802halvesd 12399 . . . . . . . . . . . . 13 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
8238sqvald 14048 . . . . . . . . . . . . 13 (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
8381, 82eqtrd 2776 . . . . . . . . . . . 12 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
8483oveq1d 7372 . . . . . . . . . . 11 (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8573recnd 11183 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
8650, 50, 85add32d 11382 . . . . . . . . . . 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 2782 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
8887oveq1d 7372 . . . . . . . . 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 11137 . . . . . . . . . . 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 7373 . . . . . . . . 9 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
9276, 88, 913eqtr4rd 2787 . . . . . . . 8 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
9392oveq1d 7372 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
9490, 85eqeltrd 2838 . . . . . . . 8 (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
9550, 94, 51addsubassd 11532 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
9642, 50, 51subsub4d 11543 . . . . . . 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 2784 . . . . . 6 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
9871, 97oveq12d 7375 . . . . 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 11561 . . . . 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 2776 . . . 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 6846 . . 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 15343 . . 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 5127 . 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 26360 . . . . . . 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 12204 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
1071, 106fsumrecl 15619 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108107, 40resubcld 11583 . . . . . . . . 9 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109108recnd 11183 . . . . . . . 8 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110109abscld 15321 . . . . . . 7 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
1112rprecred 12968 . . . . . . 7 (𝜑 → (1 / 𝐴) ∈ ℝ)
112 0red 11158 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
113 1red 11156 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
114 0lt1 11677 . . . . . . . . 9 0 < 1
115114a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
116 loge 25942 . . . . . . . . 9 (log‘e) = 1
117 mulog2sumlem1.3 . . . . . . . . . 10 (𝜑 → e ≤ 𝐴)
118 epr 16090 . . . . . . . . . . 11 e ∈ ℝ+
119 logleb 25958 . . . . . . . . . . 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 231 . . . . . . . . 9 (𝜑 → (log‘e) ≤ (log‘𝐴))
122116, 121eqbrtrrid 5141 . . . . . . . 8 (𝜑 → 1 ≤ (log‘𝐴))
123112, 113, 11, 115, 122ltletrd 11315 . . . . . . 7 (𝜑 → 0 < (log‘𝐴))
124 lemul2 12008 . . . . . . 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 1374 . . . . . 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 231 . . . . 5 (𝜑 → ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴)))
12745rpcnd 12959 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
12845rpne0d 12962 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
12963, 127, 128divrecd 11934 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130129sumeq2dv 15588 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131106recnd 11183 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
1321, 38, 131fsummulc2 15669 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133130, 132eqtr4d 2779 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134133oveq1d 7372 . . . . . . . 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 15618 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
13638, 135, 41subdid 11611 . . . . . . . 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 2779 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
138137fveq2d 6846 . . . . . 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 11512 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
14038, 139absmuld 15339 . . . . . 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 11303 . . . . . . . 8 (𝜑 → 0 ≤ (log‘𝐴))
14211, 141absidd 15307 . . . . . . 7 (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143142oveq1d 7372 . . . . . 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 2780 . . . . 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 12959 . . . . . 6 (𝜑𝐴 ∈ ℂ)
1462rpne0d 12962 . . . . . 6 (𝜑𝐴 ≠ 0)
14738, 145, 146divrecd 11934 . . . . 5 (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148126, 144, 1473brtr4d 5137 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149 fveq2 6842 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150 id 22 . . . . . . . . . . . . . 14 (𝑖 = 𝑚𝑖 = 𝑚)
151149, 150oveq12d 7375 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152151cbvsumv 15581 . . . . . . . . . . . 12 Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153 fveq2 6842 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154153oveq2d 7373 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155154sumeq1d 15586 . . . . . . . . . . . 12 (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156152, 155eqtrid 2788 . . . . . . . . . . 11 (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157 fveq2 6842 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158157oveq1d 7372 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159158oveq1d 7372 . . . . . . . . . . 11 (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160156, 159oveq12d 7375 . . . . . . . . . 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 7390 . . . . . . . . . 10 𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162160, 19, 161fvmpt 6948 . . . . . . . . 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 7372 . . . . . . 7 (𝜑 → ((𝐹𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
16549, 50, 51subsub4d 11543 . . . . . . 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 2776 . . . . . 6 (𝜑 → ((𝐹𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167166fveq2d 6846 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
16820simp3i 1141 . . . . . 6 ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
16924, 2, 117, 168syl3anc 1371 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170167, 169eqbrtrrd 5129 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ≤ ((log‘𝐴) / 𝐴))
17144, 54, 57, 57, 148, 170le2addd 11774 . . 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 11183 . . . 4 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
1731722timesd 12396 . . 3 (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174171, 173breqtrrd 5133 . 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 11312 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943   class class class wbr 5105  cmpt 5188  dom cdm 5633  wf 6492  cfv 6496  (class class class)co 7357  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  2c2 12208  +crp 12915  ...cfz 13424  cfl 13695  cexp 13967  abscabs 15119  𝑟 crli 15367  Σcsu 15570  eceu 15945  logclog 25910  γcem 26341
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-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-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-sum 15571  df-ef 15950  df-e 15951  df-sin 15952  df-cos 15953  df-tan 15954  df-pi 15955  df-dvds 16137  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
This theorem is referenced by:  mulog2sumlem2  26883
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