Theorem mulog2sumlem1 26919

Description: Asymptotic formula for Σ𝑛 ≤ 𝑥, log(𝑥 / 𝑛) / 𝑛 = (1 / 2)log↑2(𝑥) + γ · log𝑥 − 𝐿 + 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	⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿)
mulog2sumlem1.2	⊢ (𝜑 → 𝐴 ∈ 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.

Step	Hyp	Ref	Expression
1		fzfid 13888	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		mulog2sumlem1.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ+)
3		elfznn 13480	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
4	3	nnrpd 12964	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5		rpdivcl 12949	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑚 ∈ ℝ+) → (𝐴 / 𝑚) ∈ ℝ+)
6	2, 4, 5	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑚) ∈ ℝ+)
7	6	relogcld 26015	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
8	3	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
9	7, 8	nndivred 12216	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
10	1, 9	fsumrecl 15630	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
11	2	relogcld 26015	. . . . . . . 8 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
12	11	resqcld 14040	. . . . . . 7 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
13	12	rehalfcld 12409	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14		emre 26392	. . . . . . . 8 ⊢ γ ∈ ℝ
15		remulcl 11145	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
16	14, 11, 15	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17		rpsup 13781	. . . . . . . . 9 ⊢ sup(ℝ+, ℝ*, < ) = +∞
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → sup(ℝ+, ℝ*, < ) = +∞)
19		logdivsum.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))
20	19	logdivsum 26918	. . . . . . . . . . . 12 ⊢ (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
21	20	simp1i 1139	. . . . . . . . . . 11 ⊢ 𝐹:ℝ+⟶ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ+⟶ℝ)
23	22	feqmptd 6915	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹‘𝑥)))
24		mulog2sumlem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿)
25	23, 24	eqbrtrrd 5134	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹‘𝑥)) ⇝𝑟 𝐿)
26	21	ffvelcdmi 7039	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (𝐹‘𝑥) ∈ ℝ)
27	26	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ∈ ℝ)
28	18, 25, 27	rlimrecl 15474	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ)
29	16, 28	resubcld 11592	. . . . . 6 ⊢ (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
30	13, 29	readdcld 11193	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
31	10, 30	resubcld 11592	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
32	31	recnd 11192	. . 3 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
33	32	abscld 15333	. 2 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34		rerpdivcl 12954	. . . . . . . 8 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝑚 ∈ ℝ+) → ((log‘𝐴) / 𝑚) ∈ ℝ)
35	11, 4, 34	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℝ)
36	35	recnd 11192	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
37	1, 36	fsumcl 15629	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
38	11	recnd 11192	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
39		readdcl 11143	. . . . . . . 8 ⊢ (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) + γ) ∈ ℝ)
40	11, 14, 39	sylancl 586	. . . . . . 7 ⊢ (𝜑 → ((log‘𝐴) + γ) ∈ ℝ)
41	40	recnd 11192	. . . . . 6 ⊢ (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
42	38, 41	mulcld 11184	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
43	37, 42	subcld 11521	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
44	43	abscld 15333	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
45	8	nnrpd 12964	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
46	45	relogcld 26015	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
47	46, 8	nndivred 12216	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
48	47	recnd 11192	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
49	1, 48	fsumcl 15629	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
50	13	recnd 11192	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
51	28	recnd 11192	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
52	50, 51	addcld 11183	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
53	49, 52	subcld 11521	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
54	53	abscld 15333	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
55	44, 54	readdcld 11193	. 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 12236	. . 3 ⊢ 2 ∈ ℝ
57	11, 2	rerpdivcld 12997	. . 3 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58		remulcl 11145	. . 3 ⊢ ((2 ∈ ℝ ∧ ((log‘𝐴) / 𝐴) ∈ ℝ) → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
59	56, 57, 58	sylancr 587	. 2 ⊢ (𝜑 → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
60		relogdiv 25985	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑚 ∈ ℝ+) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
61	2, 4, 60	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
62	61	oveq1d 7377	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
63	38	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
64	46	recnd 11192	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
65	45	rpcnne0d 12975	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66		divsubdir 11858	. . . . . . . . . 10 ⊢ (((log‘𝐴) ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
67	63, 64, 65, 66	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
68	62, 67	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
69	68	sumeq2dv 15599	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
70	1, 36, 48	fsumsub 15684	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
71	69, 70	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72		remulcl 11145	. . . . . . . . . . . . 13 ⊢ (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) · γ) ∈ ℝ)
73	11, 14, 72	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) · γ) ∈ ℝ)
74	13, 73	readdcld 11193	. . . . . . . . . . 11 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
75	74	recnd 11192	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
76	75, 50	pncand 11522	. . . . . . . . 9 ⊢ (𝜑 → ((((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)) − (((log‘𝐴)↑2) / 2)) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
77	14	recni 11178	. . . . . . . . . . . . 13 ⊢ γ ∈ ℂ
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → γ ∈ ℂ)
79	38, 38, 78	adddid 11188	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
80	12	recnd 11192	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81	80	2halvesd 12408	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
82	38	sqvald 14058	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
83	81, 82	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
84	83	oveq1d 7377	. . . . . . . . . . 11 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
85	73	recnd 11192	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
86	50, 50, 85	add32d 11391	. . . . . . . . . . 11 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
87	79, 84, 86	3eqtr2d 2777	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
88	87	oveq1d 7377	. . . . . . . . 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 11146	. . . . . . . . . . 11 ⊢ ((γ ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
90	77, 38, 89	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
91	90	oveq2d 7378	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
92	76, 88, 91	3eqtr4rd 2782	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
93	92	oveq1d 7377	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
94	90, 85	eqeltrd 2832	. . . . . . . 8 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
95	50, 94, 51	addsubassd 11541	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
96	42, 50, 51	subsub4d 11552	. . . . . . 7 ⊢ (𝜑 → ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
97	93, 95, 96	3eqtr3d 2779	. . . . . 6 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
98	71, 97	oveq12d 7380	. . . . 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))))
99	37, 49, 42, 52	sub4d 11570	. . . . 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))))
100	98, 99	eqtrd 2771	. . . 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))))
101	100	fveq2d 6851	. . 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)))))
102	43, 53	abs2dif2d 15355	. . 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)))))
103	101, 102	eqbrtrd 5132	. 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 26397	. . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))
105	2, 104	syl 17	. . . . . 6 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))
106	8	nnrecred 12213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
107	1, 106	fsumrecl 15630	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108	107, 40	resubcld 11592	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109	108	recnd 11192	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110	109	abscld 15333	. . . . . . 7 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
111	2	rprecred 12977	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
112		0red 11167	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
113		1red 11165	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
114		0lt1 11686	. . . . . . . . 9 ⊢ 0 < 1
115	114	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 1)
116		loge 25979	. . . . . . . . 9 ⊢ (log‘e) = 1
117		mulog2sumlem1.3	. . . . . . . . . 10 ⊢ (𝜑 → e ≤ 𝐴)
118		epr 16101	. . . . . . . . . . 11 ⊢ e ∈ ℝ+
119		logleb 25995	. . . . . . . . . . 11 ⊢ ((e ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
120	118, 2, 119	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
121	117, 120	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (log‘e) ≤ (log‘𝐴))
122	116, 121	eqbrtrrid 5146	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝐴))
123	112, 113, 11, 115, 122	ltletrd 11324	. . . . . . 7 ⊢ (𝜑 → 0 < (log‘𝐴))
124		lemul2 12017	. . . . . . 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))))
125	110, 111, 11, 123, 124	syl112anc 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))))
126	105, 125	mpbid 231	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴)))
127	45	rpcnd 12968	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
128	45	rpne0d 12971	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
129	63, 127, 128	divrecd 11943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130	129	sumeq2dv 15599	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131	106	recnd 11192	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
132	1, 38, 131	fsummulc2 15680	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133	130, 132	eqtr4d 2774	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134	133	oveq1d 7377	. . . . . . . 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))))
135	1, 131	fsumcl 15629	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
136	38, 135, 41	subdid 11620	. . . . . . . 8 ⊢ (𝜑 → ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) = (((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) − ((log‘𝐴) · ((log‘𝐴) + γ))))
137	134, 136	eqtr4d 2774	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
138	137	fveq2d 6851	. . . . . 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)))))
139	135, 41	subcld 11521	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
140	38, 139	absmuld 15351	. . . . . 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)))))
141	112, 11, 123	ltled 11312	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘𝐴))
142	11, 141	absidd 15319	. . . . . . 7 ⊢ (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143	142	oveq1d 7377	. . . . . 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)))))
144	138, 140, 143	3eqtrd 2775	. . . . 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)))))
145	2	rpcnd 12968	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
146	2	rpne0d 12971	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0)
147	38, 145, 146	divrecd 11943	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148	126, 144, 147	3brtr4d 5142	. . . 4 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149		fveq2 6847	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → 𝑖 = 𝑚)
151	149, 150	oveq12d 7380	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152	151	cbvsumv 15592	. . . . . . . . . . . 12 ⊢ Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153		fveq2 6847	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154	153	oveq2d 7378	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155	154	sumeq1d 15597	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156	152, 155	eqtrid 2783	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157		fveq2 6847	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158	157	oveq1d 7377	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159	158	oveq1d 7377	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160	156, 159	oveq12d 7380	. . . . . . . . . 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 7395	. . . . . . . . . 10 ⊢ (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162	160, 19, 161	fvmpt 6953	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ+ → (𝐹‘𝐴) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)))
163	2, 162	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)))
164	163	oveq1d 7377	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
165	49, 50, 51	subsub4d 11552	. . . . . . 7 ⊢ (𝜑 → ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
166	164, 165	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167	166	fveq2d 6851	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) − 𝐿)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
168	20	simp3i 1141	. . . . . 6 ⊢ ((𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
169	24, 2, 117, 168	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170	167, 169	eqbrtrrd 5134	. . . 4 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ≤ ((log‘𝐴) / 𝐴))
171	44, 54, 57, 57, 148, 170	le2addd 11783	. . 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)))
172	57	recnd 11192	. . . 4 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
173	172	2timesd 12405	. . 3 ⊢ (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174	171, 173	breqtrrd 5138	. 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)))
175	33, 55, 59, 103, 174	letrd 11321	1 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ≤ (2 · ((log‘𝐴) / 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5638  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  supcsup 9385  ℂcc 11058  ℝcr 11059  0cc0 11060  1c1 11061   + caddc 11063   · cmul 11065  +∞cpnf 11195  ℝ^*cxr 11197   < clt 11198   ≤ cle 11199   − cmin 11394   / cdiv 11821  ℕcn 12162  2c2 12217  ℝ⁺crp 12924  ...cfz 13434  ⌊cfl 13705  ↑cexp 13977  abscabs 15131   ⇝_𝑟 crli 15379  Σcsu 15582  eceu 15956  logclog 25947  γcem 26378

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138  ax-addf 11139  ax-mulf 11140

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-fi 9356  df-sup 9387  df-inf 9388  df-oi 9455  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12423  df-xnn0 12495  df-z 12509  df-dec 12628  df-uz 12773  df-q 12883  df-rp 12925  df-xneg 13042  df-xadd 13043  df-xmul 13044  df-ioo 13278  df-ioc 13279  df-ico 13280  df-icc 13281  df-fz 13435  df-fzo 13578  df-fl 13707  df-mod 13785  df-seq 13917  df-exp 13978  df-fac 14184  df-bc 14213  df-hash 14241  df-shft 14964  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-limsup 15365  df-clim 15382  df-rlim 15383  df-sum 15583  df-ef 15961  df-e 15962  df-sin 15963  df-cos 15964  df-tan 15965  df-pi 15966  df-dvds 16148  df-struct 17030  df-sets 17047  df-slot 17065  df-ndx 17077  df-base 17095  df-ress 17124  df-plusg 17160  df-mulr 17161  df-starv 17162  df-sca 17163  df-vsca 17164  df-ip 17165  df-tset 17166  df-ple 17167  df-ds 17169  df-unif 17170  df-hom 17171  df-cco 17172  df-rest 17318  df-topn 17319  df-0g 17337  df-gsum 17338  df-topgen 17339  df-pt 17340  df-prds 17343  df-xrs 17398  df-qtop 17403  df-imas 17404  df-xps 17406  df-mre 17480  df-mrc 17481  df-acs 17483  df-mgm 18511  df-sgrp 18560  df-mnd 18571  df-submnd 18616  df-mulg 18887  df-cntz 19111  df-cmn 19578  df-psmet 20825  df-xmet 20826  df-met 20827  df-bl 20828  df-mopn 20829  df-fbas 20830  df-fg 20831  df-cnfld 20834  df-top 22280  df-topon 22297  df-topsp 22319  df-bases 22333  df-cld 22407  df-ntr 22408  df-cls 22409  df-nei 22486  df-lp 22524  df-perf 22525  df-cn 22615  df-cnp 22616  df-haus 22703  df-cmp 22775  df-tx 22950  df-hmeo 23143  df-fil 23234  df-fm 23326  df-flim 23327  df-flf 23328  df-xms 23710  df-ms 23711  df-tms 23712  df-cncf 24278  df-limc 25267  df-dv 25268  df-ulm 25773  df-log 25949  df-cxp 25950  df-atan 26254  df-em 26379

This theorem is referenced by:  mulog2sumlem2  26920