MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulog2sumlem1 Structured version   Visualization version   GIF version

Theorem mulog2sumlem1 25268
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 12812 . . . . . 6 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 mulog2sumlem1.2 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ+)
3 elfznn 12408 . . . . . . . . . 10 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
43nnrpd 11908 . . . . . . . . 9 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5 rpdivcl 11894 . . . . . . . . 9 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (𝐴 / 𝑚) ∈ ℝ+)
62, 4, 5syl2an 493 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑚) ∈ ℝ+)
76relogcld 24414 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
83adantl 481 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
97, 8nndivred 11107 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
101, 9fsumrecl 14509 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
112relogcld 24414 . . . . . . . 8 (𝜑 → (log‘𝐴) ∈ ℝ)
1211resqcld 13075 . . . . . . 7 (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
1312rehalfcld 11317 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14 emre 24777 . . . . . . . 8 γ ∈ ℝ
15 remulcl 10059 . . . . . . . 8 ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
1614, 11, 15sylancr 696 . . . . . . 7 (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17 rpsup 12705 . . . . . . . . 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 25267 . . . . . . . . . . . 12 (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
2120simp1i 1090 . . . . . . . . . . 11 𝐹:ℝ+⟶ℝ
2221a1i 11 . . . . . . . . . 10 (𝜑𝐹:ℝ+⟶ℝ)
2322feqmptd 6288 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)))
24 mulog2sumlem.1 . . . . . . . . 9 (𝜑𝐹𝑟 𝐿)
2523, 24eqbrtrrd 4709 . . . . . . . 8 (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹𝑥)) ⇝𝑟 𝐿)
2621ffvelrni 6398 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝐹𝑥) ∈ ℝ)
2726adantl 481 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ+) → (𝐹𝑥) ∈ ℝ)
2818, 25, 27rlimrecl 14355 . . . . . . 7 (𝜑𝐿 ∈ ℝ)
2916, 28resubcld 10496 . . . . . 6 (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
3013, 29readdcld 10107 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
3110, 30resubcld 10496 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
3231recnd 10106 . . 3 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
3332abscld 14219 . 2 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34 rerpdivcl 11899 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ 𝑚 ∈ ℝ+) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3511, 4, 34syl2an 493 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℝ)
3635recnd 10106 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
371, 36fsumcl 14508 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
3811recnd 10106 . . . . . 6 (𝜑 → (log‘𝐴) ∈ ℂ)
39 readdcl 10057 . . . . . . . 8 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) + γ) ∈ ℝ)
4011, 14, 39sylancl 695 . . . . . . 7 (𝜑 → ((log‘𝐴) + γ) ∈ ℝ)
4140recnd 10106 . . . . . 6 (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
4238, 41mulcld 10098 . . . . 5 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
4337, 42subcld 10430 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
4443abscld 14219 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
458nnrpd 11908 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
4645relogcld 24414 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
4746, 8nndivred 11107 . . . . . . 7 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
4847recnd 10106 . . . . . 6 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
491, 48fsumcl 14508 . . . . 5 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
5013recnd 10106 . . . . . 6 (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
5128recnd 10106 . . . . . 6 (𝜑𝐿 ∈ ℂ)
5250, 51addcld 10097 . . . . 5 (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
5349, 52subcld 10430 . . . 4 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
5453abscld 14219 . . 3 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
5544, 54readdcld 10107 . 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 11128 . . 3 2 ∈ ℝ
5711, 2rerpdivcld 11941 . . 3 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58 remulcl 10059 . . 3 ((2 ∈ ℝ ∧ ((log‘𝐴) / 𝐴) ∈ ℝ) → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
5956, 57, 58sylancr 696 . 2 (𝜑 → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
60 relogdiv 24384 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+𝑚 ∈ ℝ+) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
612, 4, 60syl2an 493 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
6261oveq1d 6705 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
6338adantr 480 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
6446recnd 10106 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
6545rpcnne0d 11919 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66 divsubdir 10759 . . . . . . . . . 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 1366 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6862, 67eqtrd 2685 . . . . . . . 8 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
6968sumeq2dv 14477 . . . . . . 7 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
701, 36, 48fsumsub 14564 . . . . . . 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 2685 . . . . . 6 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72 remulcl 10059 . . . . . . . . . . . . 13 (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) · γ) ∈ ℝ)
7311, 14, 72sylancl 695 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℝ)
7413, 73readdcld 10107 . . . . . . . . . . 11 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
7574recnd 10106 . . . . . . . . . 10 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
7675, 50pncand 10431 . . . . . . . . 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 10090 . . . . . . . . . . . . 13 γ ∈ ℂ
7877a1i 11 . . . . . . . . . . . 12 (𝜑 → γ ∈ ℂ)
7938, 38, 78adddid 10102 . . . . . . . . . . 11 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8012recnd 10106 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81802halvesd 11316 . . . . . . . . . . . . 13 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
8238sqvald 13045 . . . . . . . . . . . . 13 (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
8381, 82eqtrd 2685 . . . . . . . . . . . 12 (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
8483oveq1d 6705 . . . . . . . . . . 11 (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
8573recnd 10106 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
8650, 50, 85add32d 10301 . . . . . . . . . . 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 2691 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
8887oveq1d 6705 . . . . . . . . 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 10060 . . . . . . . . . . 11 ((γ ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
9077, 38, 89sylancr 696 . . . . . . . . . 10 (𝜑 → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
9190oveq2d 6706 . . . . . . . . 9 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
9276, 88, 913eqtr4rd 2696 . . . . . . . 8 (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
9392oveq1d 6705 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
9490, 85eqeltrd 2730 . . . . . . . 8 (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
9550, 94, 51addsubassd 10450 . . . . . . 7 (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
9642, 50, 51subsub4d 10461 . . . . . . 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 2693 . . . . . 6 (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
9871, 97oveq12d 6708 . . . . 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 10479 . . . . 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 2685 . . . 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 6233 . . 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 14241 . . 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 4707 . 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 24782 . . . . . . 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 11104 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
1071, 106fsumrecl 14509 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108107, 40resubcld 10496 . . . . . . . . 9 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109108recnd 10106 . . . . . . . 8 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110109abscld 14219 . . . . . . 7 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
1112rprecred 11921 . . . . . . 7 (𝜑 → (1 / 𝐴) ∈ ℝ)
112 0red 10079 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
113 1red 10093 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
114 0lt1 10588 . . . . . . . . 9 0 < 1
115114a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
116 loge 24378 . . . . . . . . 9 (log‘e) = 1
117 mulog2sumlem1.3 . . . . . . . . . 10 (𝜑 → e ≤ 𝐴)
118 epr 14980 . . . . . . . . . . 11 e ∈ ℝ+
119 logleb 24394 . . . . . . . . . . 11 ((e ∈ ℝ+𝐴 ∈ ℝ+) → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
120118, 2, 119sylancr 696 . . . . . . . . . 10 (𝜑 → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
121117, 120mpbid 222 . . . . . . . . 9 (𝜑 → (log‘e) ≤ (log‘𝐴))
122116, 121syl5eqbrr 4721 . . . . . . . 8 (𝜑 → 1 ≤ (log‘𝐴))
123112, 113, 11, 115, 122ltletrd 10235 . . . . . . 7 (𝜑 → 0 < (log‘𝐴))
124 lemul2 10914 . . . . . . 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 1370 . . . . . 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 11912 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
12845rpne0d 11915 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
12963, 127, 128divrecd 10842 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130129sumeq2dv 14477 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131106recnd 10106 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
1321, 38, 131fsummulc2 14560 . . . . . . . . . 10 (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133130, 132eqtr4d 2688 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134133oveq1d 6705 . . . . . . . 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 14508 . . . . . . . . 9 (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
13638, 135, 41subdid 10524 . . . . . . . 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 2688 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
138137fveq2d 6233 . . . . . 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 10430 . . . . . . 7 (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
14038, 139absmuld 14237 . . . . . 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 10223 . . . . . . . 8 (𝜑 → 0 ≤ (log‘𝐴))
14211, 141absidd 14205 . . . . . . 7 (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143142oveq1d 6705 . . . . . 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 2689 . . . . 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 11912 . . . . . 6 (𝜑𝐴 ∈ ℂ)
1462rpne0d 11915 . . . . . 6 (𝜑𝐴 ≠ 0)
14738, 145, 146divrecd 10842 . . . . 5 (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148126, 144, 1473brtr4d 4717 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149 fveq2 6229 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150 id 22 . . . . . . . . . . . . . 14 (𝑖 = 𝑚𝑖 = 𝑚)
151149, 150oveq12d 6708 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152151cbvsumv 14470 . . . . . . . . . . . 12 Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153 fveq2 6229 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154153oveq2d 6706 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155154sumeq1d 14475 . . . . . . . . . . . 12 (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156152, 155syl5eq 2697 . . . . . . . . . . 11 (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157 fveq2 6229 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158157oveq1d 6705 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159158oveq1d 6705 . . . . . . . . . . 11 (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160156, 159oveq12d 6708 . . . . . . . . . 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 6718 . . . . . . . . . 10 𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162160, 19, 161fvmpt 6321 . . . . . . . . 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 6705 . . . . . . 7 (𝜑 → ((𝐹𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
16549, 50, 51subsub4d 10461 . . . . . . 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 2685 . . . . . 6 (𝜑 → ((𝐹𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167166fveq2d 6233 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
16820simp3i 1092 . . . . . 6 ((𝐹𝑟 𝐿𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
16924, 2, 117, 168syl3anc 1366 . . . . 5 (𝜑 → (abs‘((𝐹𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170167, 169eqbrtrrd 4709 . . . 4 (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ≤ ((log‘𝐴) / 𝐴))
17144, 54, 57, 57, 148, 170le2addd 10684 . . 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 10106 . . . 4 (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
1731722timesd 11313 . . 3 (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174171, 173breqtrrd 4713 . 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 10232 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cmpt 4762  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  +crp 11870  ...cfz 12364  cfl 12631  cexp 12900  abscabs 14018  𝑟 crli 14260  Σcsu 14460  eceu 14837  logclog 24346  γcem 24763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-e 14843  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-cxp 24349  df-em 24764
This theorem is referenced by:  mulog2sumlem2  25269
  Copyright terms: Public domain W3C validator