Theorem mulog2sumlem1 27502

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 13996	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		mulog2sumlem1.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ+)
3		elfznn 13575	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
4	3	nnrpd 13054	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5		rpdivcl 13039	. . . . . . . . 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 26589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
8	3	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
9	7, 8	nndivred 12299	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
10	1, 9	fsumrecl 15755	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
11	2	relogcld 26589	. . . . . . . 8 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
12	11	resqcld 14148	. . . . . . 7 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
13	12	rehalfcld 12493	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14		emre 26973	. . . . . . . 8 ⊢ γ ∈ ℝ
15		remulcl 11219	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
16	14, 11, 15	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17		rpsup 13888	. . . . . . . . 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 27501	. . . . . . . . . . . 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 6952	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹‘𝑥)))
24		mulog2sumlem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿)
25	23, 24	eqbrtrrd 5148	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹‘𝑥)) ⇝𝑟 𝐿)
26	21	ffvelcdmi 7078	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (𝐹‘𝑥) ∈ ℝ)
27	26	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ∈ ℝ)
28	18, 25, 27	rlimrecl 15601	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ)
29	16, 28	resubcld 11670	. . . . . 6 ⊢ (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
30	13, 29	readdcld 11269	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
31	10, 30	resubcld 11670	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
32	31	recnd 11268	. . 3 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
33	32	abscld 15460	. 2 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34		rerpdivcl 13044	. . . . . . . 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 11268	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
37	1, 36	fsumcl 15754	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
38	11	recnd 11268	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
39		readdcl 11217	. . . . . . . 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 11268	. . . . . 6 ⊢ (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
42	38, 41	mulcld 11260	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
43	37, 42	subcld 11599	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
44	43	abscld 15460	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
45	8	nnrpd 13054	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
46	45	relogcld 26589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
47	46, 8	nndivred 12299	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
48	47	recnd 11268	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
49	1, 48	fsumcl 15754	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
50	13	recnd 11268	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
51	28	recnd 11268	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
52	50, 51	addcld 11259	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
53	49, 52	subcld 11599	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
54	53	abscld 15460	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
55	44, 54	readdcld 11269	. 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 12319	. . 3 ⊢ 2 ∈ ℝ
57	11, 2	rerpdivcld 13087	. . 3 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58		remulcl 11219	. . 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 26559	. . . . . . . . . . 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 7425	. . . . . . . . 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 11268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
65	45	rpcnne0d 13065	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66		divsubdir 11940	. . . . . . . . . 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 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
69	68	sumeq2dv 15723	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
70	1, 36, 48	fsumsub 15809	. . . . . . 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 11219	. . . . . . . . . . . . 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 11269	. . . . . . . . . . 11 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
75	74	recnd 11268	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
76	75, 50	pncand 11600	. . . . . . . . 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 11254	. . . . . . . . . . . . 13 ⊢ γ ∈ ℂ
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → γ ∈ ℂ)
79	38, 38, 78	adddid 11264	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
80	12	recnd 11268	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81	80	2halvesd 12492	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
82	38	sqvald 14166	. . . . . . . . . . . . 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 7425	. . . . . . . . . . 11 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
85	73	recnd 11268	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
86	50, 50, 85	add32d 11468	. . . . . . . . . . 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 7425	. . . . . . . . 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 11220	. . . . . . . . . . 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 7426	. . . . . . . . 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 7425	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
94	90, 85	eqeltrd 2835	. . . . . . . 8 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
95	50, 94, 51	addsubassd 11619	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
96	42, 50, 51	subsub4d 11630	. . . . . . 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 7428	. . . . 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 11648	. . . . 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 6885	. . 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 15482	. . 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 5146	. 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 26978	. . . . . . 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 12296	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
107	1, 106	fsumrecl 15755	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108	107, 40	resubcld 11670	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109	108	recnd 11268	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110	109	abscld 15460	. . . . . . 7 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
111	2	rprecred 13067	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
112		0red 11243	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
113		1red 11241	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
114		0lt1 11764	. . . . . . . . 9 ⊢ 0 < 1
115	114	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 1)
116		loge 26552	. . . . . . . . 9 ⊢ (log‘e) = 1
117		mulog2sumlem1.3	. . . . . . . . . 10 ⊢ (𝜑 → e ≤ 𝐴)
118		epr 16231	. . . . . . . . . . 11 ⊢ e ∈ ℝ+
119		logleb 26569	. . . . . . . . . . 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 5160	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝐴))
123	112, 113, 11, 115, 122	ltletrd 11400	. . . . . . 7 ⊢ (𝜑 → 0 < (log‘𝐴))
124		lemul2 12099	. . . . . . 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 13058	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
128	45	rpne0d 13061	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
129	63, 127, 128	divrecd 12025	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130	129	sumeq2dv 15723	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131	106	recnd 11268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
132	1, 38, 131	fsummulc2 15805	. . . . . . . . . 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 7425	. . . . . . . 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 15754	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
136	38, 135, 41	subdid 11698	. . . . . . . 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 6885	. . . . . 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 11599	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
140	38, 139	absmuld 15478	. . . . . 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 11388	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘𝐴))
142	11, 141	absidd 15446	. . . . . . 7 ⊢ (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143	142	oveq1d 7425	. . . . . 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 13058	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
146	2	rpne0d 13061	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0)
147	38, 145, 146	divrecd 12025	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148	126, 144, 147	3brtr4d 5156	. . . 4 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → 𝑖 = 𝑚)
151	149, 150	oveq12d 7428	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152	151	cbvsumv 15717	. . . . . . . . . . . 12 ⊢ Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154	153	oveq2d 7426	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155	154	sumeq1d 15721	. . . . . . . . . . . 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 6881	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158	157	oveq1d 7425	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159	158	oveq1d 7425	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160	156, 159	oveq12d 7428	. . . . . . . . . 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 7443	. . . . . . . . . 10 ⊢ (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162	160, 19, 161	fvmpt 6991	. . . . . . . . 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 7425	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
165	49, 50, 51	subsub4d 11630	. . . . . . 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 6885	. . . . 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 5148	. . . 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 11861	. . 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 11268	. . . 4 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
173	172	2timesd 12489	. . 3 ⊢ (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174	171, 173	breqtrrd 5152	. 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 11397	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 1540   ∈ wcel 2109   ≠ wne 2933   class class class wbr 5124   ↦ cmpt 5206  dom cdm 5659  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  supcsup 9457  ℂcc 11132  ℝcr 11133  0cc0 11134  1c1 11135   + caddc 11137   · cmul 11139  +∞cpnf 11271  ℝ^*cxr 11273   < clt 11274   ≤ cle 11275   − cmin 11471   / cdiv 11899  ℕcn 12245  2c2 12300  ℝ⁺crp 13013  ...cfz 13529  ⌊cfl 13812  ↑cexp 14084  abscabs 15258   ⇝_𝑟 crli 15506  Σcsu 15707  eceu 16083  logclog 26520  γcem 26959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212  ax-addf 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8724  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-fi 9428  df-sup 9459  df-inf 9460  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-xnn0 12580  df-z 12594  df-dec 12714  df-uz 12858  df-q 12970  df-rp 13014  df-xneg 13133  df-xadd 13134  df-xmul 13135  df-ioo 13371  df-ioc 13372  df-ico 13373  df-icc 13374  df-fz 13530  df-fzo 13677  df-fl 13814  df-mod 13892  df-seq 14025  df-exp 14085  df-fac 14297  df-bc 14326  df-hash 14354  df-shft 15091  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-limsup 15492  df-clim 15509  df-rlim 15510  df-sum 15708  df-ef 16088  df-e 16089  df-sin 16090  df-cos 16091  df-tan 16092  df-pi 16093  df-dvds 16278  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-starv 17291  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-unif 17299  df-hom 17300  df-cco 17301  df-rest 17441  df-topn 17442  df-0g 17460  df-gsum 17461  df-topgen 17462  df-pt 17463  df-prds 17466  df-xrs 17521  df-qtop 17526  df-imas 17527  df-xps 17529  df-mre 17603  df-mrc 17604  df-acs 17606  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-submnd 18767  df-mulg 19056  df-cntz 19305  df-cmn 19768  df-psmet 21312  df-xmet 21313  df-met 21314  df-bl 21315  df-mopn 21316  df-fbas 21317  df-fg 21318  df-cnfld 21321  df-top 22837  df-topon 22854  df-topsp 22876  df-bases 22889  df-cld 22962  df-ntr 22963  df-cls 22964  df-nei 23041  df-lp 23079  df-perf 23080  df-cn 23170  df-cnp 23171  df-haus 23258  df-cmp 23330  df-tx 23505  df-hmeo 23698  df-fil 23789  df-fm 23881  df-flim 23882  df-flf 23883  df-xms 24264  df-ms 24265  df-tms 24266  df-cncf 24827  df-limc 25824  df-dv 25825  df-ulm 26343  df-log 26522  df-cxp 26523  df-atan 26834  df-em 26960

This theorem is referenced by:  mulog2sumlem2  27503