Theorem mulog2sumlem1 27482

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 13890	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		mulog2sumlem1.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ+)
3		elfznn 13463	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
4	3	nnrpd 12942	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5		rpdivcl 12927	. . . . . . . . 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 26569	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
8	3	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
9	7, 8	nndivred 12189	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
10	1, 9	fsumrecl 15651	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
11	2	relogcld 26569	. . . . . . . 8 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
12	11	resqcld 14042	. . . . . . 7 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
13	12	rehalfcld 12378	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14		emre 26953	. . . . . . . 8 ⊢ γ ∈ ℝ
15		remulcl 11101	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
16	14, 11, 15	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17		rpsup 13780	. . . . . . . . 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 27481	. . . . . . . . . . . 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 6899	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹‘𝑥)))
24		mulog2sumlem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿)
25	23, 24	eqbrtrrd 5119	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹‘𝑥)) ⇝𝑟 𝐿)
26	21	ffvelcdmi 7025	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (𝐹‘𝑥) ∈ ℝ)
27	26	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ∈ ℝ)
28	18, 25, 27	rlimrecl 15497	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ)
29	16, 28	resubcld 11555	. . . . . 6 ⊢ (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
30	13, 29	readdcld 11151	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
31	10, 30	resubcld 11555	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
32	31	recnd 11150	. . 3 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
33	32	abscld 15356	. 2 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34		rerpdivcl 12932	. . . . . . . 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 11150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
37	1, 36	fsumcl 15650	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
38	11	recnd 11150	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
39		readdcl 11099	. . . . . . . 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 11150	. . . . . 6 ⊢ (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
42	38, 41	mulcld 11142	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
43	37, 42	subcld 11482	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
44	43	abscld 15356	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
45	8	nnrpd 12942	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
46	45	relogcld 26569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
47	46, 8	nndivred 12189	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
48	47	recnd 11150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
49	1, 48	fsumcl 15650	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
50	13	recnd 11150	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
51	28	recnd 11150	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
52	50, 51	addcld 11141	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
53	49, 52	subcld 11482	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
54	53	abscld 15356	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
55	44, 54	readdcld 11151	. 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 12209	. . 3 ⊢ 2 ∈ ℝ
57	11, 2	rerpdivcld 12975	. . 3 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58		remulcl 11101	. . 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 26539	. . . . . . . . . . 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 7370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
63	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
64	46	recnd 11150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
65	45	rpcnne0d 12953	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66		divsubdir 11825	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
68	62, 67	eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
69	68	sumeq2dv 15619	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
70	1, 36, 48	fsumsub 15705	. . . . . . 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 2768	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72		remulcl 11101	. . . . . . . . . . . . 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 11151	. . . . . . . . . . 11 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
75	74	recnd 11150	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
76	75, 50	pncand 11483	. . . . . . . . 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 11136	. . . . . . . . . . . . 13 ⊢ γ ∈ ℂ
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → γ ∈ ℂ)
79	38, 38, 78	adddid 11146	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
80	12	recnd 11150	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81	80	2halvesd 12377	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
82	38	sqvald 14060	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
83	81, 82	eqtrd 2768	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
84	83	oveq1d 7370	. . . . . . . . . . 11 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
85	73	recnd 11150	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
86	50, 50, 85	add32d 11351	. . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
88	87	oveq1d 7370	. . . . . . . . 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 11102	. . . . . . . . . . 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 7371	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
92	76, 88, 91	3eqtr4rd 2779	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
93	92	oveq1d 7370	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
94	90, 85	eqeltrd 2833	. . . . . . . 8 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
95	50, 94, 51	addsubassd 11502	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
96	42, 50, 51	subsub4d 11513	. . . . . . 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 2776	. . . . . 6 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
98	71, 97	oveq12d 7373	. . . . 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 11531	. . . . 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 2768	. . . 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 6835	. . 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 15378	. . 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 5117	. 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 26958	. . . . . . 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 12186	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
107	1, 106	fsumrecl 15651	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108	107, 40	resubcld 11555	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109	108	recnd 11150	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110	109	abscld 15356	. . . . . . 7 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
111	2	rprecred 12955	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
112		0red 11125	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
113		1red 11123	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
114		0lt1 11649	. . . . . . . . 9 ⊢ 0 < 1
115	114	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 1)
116		loge 26532	. . . . . . . . 9 ⊢ (log‘e) = 1
117		mulog2sumlem1.3	. . . . . . . . . 10 ⊢ (𝜑 → e ≤ 𝐴)
118		epr 16127	. . . . . . . . . . 11 ⊢ e ∈ ℝ+
119		logleb 26549	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → (log‘e) ≤ (log‘𝐴))
122	116, 121	eqbrtrrid 5131	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝐴))
123	112, 113, 11, 115, 122	ltletrd 11283	. . . . . . 7 ⊢ (𝜑 → 0 < (log‘𝐴))
124		lemul2 11984	. . . . . . 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 1376	. . . . . 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 232	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴)))
127	45	rpcnd 12946	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
128	45	rpne0d 12949	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
129	63, 127, 128	divrecd 11910	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130	129	sumeq2dv 15619	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131	106	recnd 11150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
132	1, 38, 131	fsummulc2 15701	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133	130, 132	eqtr4d 2771	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134	133	oveq1d 7370	. . . . . . . 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 15650	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
136	38, 135, 41	subdid 11583	. . . . . . . 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 2771	. . . . . . 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 6835	. . . . . 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 11482	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
140	38, 139	absmuld 15374	. . . . . 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 11271	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘𝐴))
142	11, 141	absidd 15340	. . . . . . 7 ⊢ (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143	142	oveq1d 7370	. . . . . 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 2772	. . . . 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 12946	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
146	2	rpne0d 12949	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0)
147	38, 145, 146	divrecd 11910	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148	126, 144, 147	3brtr4d 5127	. . . 4 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149		fveq2 6831	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → 𝑖 = 𝑚)
151	149, 150	oveq12d 7373	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152	151	cbvsumv 15613	. . . . . . . . . . . 12 ⊢ Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153		fveq2 6831	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154	153	oveq2d 7371	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155	154	sumeq1d 15617	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156	152, 155	eqtrid 2780	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158	157	oveq1d 7370	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159	158	oveq1d 7370	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160	156, 159	oveq12d 7373	. . . . . . . . . 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 7388	. . . . . . . . . 10 ⊢ (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162	160, 19, 161	fvmpt 6938	. . . . . . . . 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 7370	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
165	49, 50, 51	subsub4d 11513	. . . . . . 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 2768	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167	166	fveq2d 6835	. . . . 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 1373	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170	167, 169	eqbrtrrd 5119	. . . 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 11746	. . 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 11150	. . . 4 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
173	172	2timesd 12374	. . 3 ⊢ (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174	171, 173	breqtrrd 5123	. 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 11280	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930   class class class wbr 5095   ↦ cmpt 5176  dom cdm 5621  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  supcsup 9334  ℂcc 11014  ℝcr 11015  0cc0 11016  1c1 11017   + caddc 11019   · cmul 11021  +∞cpnf 11153  ℝ^*cxr 11155   < clt 11156   ≤ cle 11157   − cmin 11354   / cdiv 11784  ℕcn 12135  2c2 12190  ℝ⁺crp 12900  ...cfz 13417  ⌊cfl 13704  ↑cexp 13978  abscabs 15151   ⇝_𝑟 crli 15402  Σcsu 15603  eceu 15979  logclog 26500  γcem 26939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9541  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093  ax-pre-sup 11094  ax-addf 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-om 7806  df-1st 7930  df-2nd 7931  df-supp 8100  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-oadd 8398  df-er 8631  df-map 8761  df-pm 8762  df-ixp 8831  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-fsupp 9256  df-fi 9305  df-sup 9336  df-inf 9337  df-oi 9406  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-div 11785  df-nn 12136  df-2 12198  df-3 12199  df-4 12200  df-5 12201  df-6 12202  df-7 12203  df-8 12204  df-9 12205  df-n0 12392  df-xnn0 12465  df-z 12479  df-dec 12599  df-uz 12743  df-q 12857  df-rp 12901  df-xneg 13021  df-xadd 13022  df-xmul 13023  df-ioo 13259  df-ioc 13260  df-ico 13261  df-icc 13262  df-fz 13418  df-fzo 13565  df-fl 13706  df-mod 13784  df-seq 13919  df-exp 13979  df-fac 14191  df-bc 14220  df-hash 14248  df-shft 14984  df-cj 15016  df-re 15017  df-im 15018  df-sqrt 15152  df-abs 15153  df-limsup 15388  df-clim 15405  df-rlim 15406  df-sum 15604  df-ef 15984  df-e 15985  df-sin 15986  df-cos 15987  df-tan 15988  df-pi 15989  df-dvds 16174  df-struct 17068  df-sets 17085  df-slot 17103  df-ndx 17115  df-base 17131  df-ress 17152  df-plusg 17184  df-mulr 17185  df-starv 17186  df-sca 17187  df-vsca 17188  df-ip 17189  df-tset 17190  df-ple 17191  df-ds 17193  df-unif 17194  df-hom 17195  df-cco 17196  df-rest 17336  df-topn 17337  df-0g 17355  df-gsum 17356  df-topgen 17357  df-pt 17358  df-prds 17361  df-xrs 17416  df-qtop 17421  df-imas 17422  df-xps 17424  df-mre 17498  df-mrc 17499  df-acs 17501  df-mgm 18558  df-sgrp 18637  df-mnd 18653  df-submnd 18702  df-mulg 18991  df-cntz 19239  df-cmn 19704  df-psmet 21293  df-xmet 21294  df-met 21295  df-bl 21296  df-mopn 21297  df-fbas 21298  df-fg 21299  df-cnfld 21302  df-top 22819  df-topon 22836  df-topsp 22858  df-bases 22871  df-cld 22944  df-ntr 22945  df-cls 22946  df-nei 23023  df-lp 23061  df-perf 23062  df-cn 23152  df-cnp 23153  df-haus 23240  df-cmp 23312  df-tx 23487  df-hmeo 23680  df-fil 23771  df-fm 23863  df-flim 23864  df-flf 23865  df-xms 24245  df-ms 24246  df-tms 24247  df-cncf 24808  df-limc 25804  df-dv 25805  df-ulm 26323  df-log 26502  df-cxp 26503  df-atan 26814  df-em 26940

This theorem is referenced by:  mulog2sumlem2  27483