Theorem mulog2sumlem1 26118

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 13336	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		mulog2sumlem1.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ+)
3		elfznn 12931	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
4	3	nnrpd 12417	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ+)
5		rpdivcl 12402	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑚 ∈ ℝ+) → (𝐴 / 𝑚) ∈ ℝ+)
6	2, 4, 5	syl2an 598	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑚) ∈ ℝ+)
7	6	relogcld 25214	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
8	3	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
9	7, 8	nndivred 11679	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
10	1, 9	fsumrecl 15083	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
11	2	relogcld 25214	. . . . . . . 8 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
12	11	resqcld 13607	. . . . . . 7 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
13	12	rehalfcld 11872	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14		emre 25591	. . . . . . . 8 ⊢ γ ∈ ℝ
15		remulcl 10611	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
16	14, 11, 15	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17		rpsup 13229	. . . . . . . . 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 26117	. . . . . . . . . . . 12 ⊢ (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
21	20	simp1i 1136	. . . . . . . . . . 11 ⊢ 𝐹:ℝ+⟶ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ+⟶ℝ)
23	22	feqmptd 6708	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ+ ↦ (𝐹‘𝑥)))
24		mulog2sumlem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿)
25	23, 24	eqbrtrrd 5054	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐹‘𝑥)) ⇝𝑟 𝐿)
26	21	ffvelrni 6827	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (𝐹‘𝑥) ∈ ℝ)
27	26	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ∈ ℝ)
28	18, 25, 27	rlimrecl 14929	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ)
29	16, 28	resubcld 11057	. . . . . 6 ⊢ (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
30	13, 29	readdcld 10659	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
31	10, 30	resubcld 11057	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
32	31	recnd 10658	. . 3 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
33	32	abscld 14788	. 2 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34		rerpdivcl 12407	. . . . . . . 8 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝑚 ∈ ℝ+) → ((log‘𝐴) / 𝑚) ∈ ℝ)
35	11, 4, 34	syl2an 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℝ)
36	35	recnd 10658	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
37	1, 36	fsumcl 15082	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
38	11	recnd 10658	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
39		readdcl 10609	. . . . . . . 8 ⊢ (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) + γ) ∈ ℝ)
40	11, 14, 39	sylancl 589	. . . . . . 7 ⊢ (𝜑 → ((log‘𝐴) + γ) ∈ ℝ)
41	40	recnd 10658	. . . . . 6 ⊢ (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
42	38, 41	mulcld 10650	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
43	37, 42	subcld 10986	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
44	43	abscld 14788	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
45	8	nnrpd 12417	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ+)
46	45	relogcld 25214	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
47	46, 8	nndivred 11679	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
48	47	recnd 10658	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
49	1, 48	fsumcl 15082	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
50	13	recnd 10658	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
51	28	recnd 10658	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
52	50, 51	addcld 10649	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
53	49, 52	subcld 10986	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
54	53	abscld 14788	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
55	44, 54	readdcld 10659	. 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 11699	. . 3 ⊢ 2 ∈ ℝ
57	11, 2	rerpdivcld 12450	. . 3 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58		remulcl 10611	. . 3 ⊢ ((2 ∈ ℝ ∧ ((log‘𝐴) / 𝐴) ∈ ℝ) → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
59	56, 57, 58	sylancr 590	. 2 ⊢ (𝜑 → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
60		relogdiv 25184	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑚 ∈ ℝ+) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
61	2, 4, 60	syl2an 598	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
62	61	oveq1d 7150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
63	38	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
64	46	recnd 10658	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
65	45	rpcnne0d 12428	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66		divsubdir 11323	. . . . . . . . . 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 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
68	62, 67	eqtrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
69	68	sumeq2dv 15052	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
70	1, 36, 48	fsumsub 15135	. . . . . . 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 2833	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72		remulcl 10611	. . . . . . . . . . . . 13 ⊢ (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) · γ) ∈ ℝ)
73	11, 14, 72	sylancl 589	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) · γ) ∈ ℝ)
74	13, 73	readdcld 10659	. . . . . . . . . . 11 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
75	74	recnd 10658	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
76	75, 50	pncand 10987	. . . . . . . . 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 10644	. . . . . . . . . . . . 13 ⊢ γ ∈ ℂ
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → γ ∈ ℂ)
79	38, 38, 78	adddid 10654	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
80	12	recnd 10658	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81	80	2halvesd 11871	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
82	38	sqvald 13503	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
83	81, 82	eqtrd 2833	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
84	83	oveq1d 7150	. . . . . . . . . . 11 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
85	73	recnd 10658	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
86	50, 50, 85	add32d 10856	. . . . . . . . . . 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 2839	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
88	87	oveq1d 7150	. . . . . . . . 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 10612	. . . . . . . . . . 11 ⊢ ((γ ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
90	77, 38, 89	sylancr 590	. . . . . . . . . 10 ⊢ (𝜑 → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
91	90	oveq2d 7151	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
92	76, 88, 91	3eqtr4rd 2844	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
93	92	oveq1d 7150	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
94	90, 85	eqeltrd 2890	. . . . . . . 8 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
95	50, 94, 51	addsubassd 11006	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
96	42, 50, 51	subsub4d 11017	. . . . . . 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 2841	. . . . . 6 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
98	71, 97	oveq12d 7153	. . . . 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 11035	. . . . 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 2833	. . . 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 6649	. . 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 14810	. . 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 5052	. 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 25596	. . . . . . 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 11676	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
107	1, 106	fsumrecl 15083	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108	107, 40	resubcld 11057	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109	108	recnd 10658	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110	109	abscld 14788	. . . . . . 7 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
111	2	rprecred 12430	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
112		0red 10633	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
113		1red 10631	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
114		0lt1 11151	. . . . . . . . 9 ⊢ 0 < 1
115	114	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 1)
116		loge 25178	. . . . . . . . 9 ⊢ (log‘e) = 1
117		mulog2sumlem1.3	. . . . . . . . . 10 ⊢ (𝜑 → e ≤ 𝐴)
118		epr 15553	. . . . . . . . . . 11 ⊢ e ∈ ℝ+
119		logleb 25194	. . . . . . . . . . 11 ⊢ ((e ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
120	118, 2, 119	sylancr 590	. . . . . . . . . 10 ⊢ (𝜑 → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
121	117, 120	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (log‘e) ≤ (log‘𝐴))
122	116, 121	eqbrtrrid 5066	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝐴))
123	112, 113, 11, 115, 122	ltletrd 10789	. . . . . . 7 ⊢ (𝜑 → 0 < (log‘𝐴))
124		lemul2 11482	. . . . . . 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 1371	. . . . . 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 235	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴)))
127	45	rpcnd 12421	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
128	45	rpne0d 12424	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
129	63, 127, 128	divrecd 11408	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130	129	sumeq2dv 15052	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131	106	recnd 10658	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
132	1, 38, 131	fsummulc2 15131	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133	130, 132	eqtr4d 2836	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134	133	oveq1d 7150	. . . . . . . 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 15082	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
136	38, 135, 41	subdid 11085	. . . . . . . 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 2836	. . . . . . 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 6649	. . . . . 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 10986	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
140	38, 139	absmuld 14806	. . . . . 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 10777	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘𝐴))
142	11, 141	absidd 14774	. . . . . . 7 ⊢ (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143	142	oveq1d 7150	. . . . . 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 2837	. . . . 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 12421	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
146	2	rpne0d 12424	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0)
147	38, 145, 146	divrecd 11408	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148	126, 144, 147	3brtr4d 5062	. . . 4 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149		fveq2 6645	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → 𝑖 = 𝑚)
151	149, 150	oveq12d 7153	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152	151	cbvsumv 15045	. . . . . . . . . . . 12 ⊢ Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153		fveq2 6645	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154	153	oveq2d 7151	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155	154	sumeq1d 15050	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156	152, 155	syl5eq 2845	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158	157	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159	158	oveq1d 7150	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160	156, 159	oveq12d 7153	. . . . . . . . . 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 7168	. . . . . . . . . 10 ⊢ (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162	160, 19, 161	fvmpt 6745	. . . . . . . . 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 7150	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
165	49, 50, 51	subsub4d 11017	. . . . . . 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 2833	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167	166	fveq2d 6649	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) − 𝐿)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
168	20	simp3i 1138	. . . . . 6 ⊢ ((𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
169	24, 2, 117, 168	syl3anc 1368	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170	167, 169	eqbrtrrd 5054	. . . 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 11248	. . 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 10658	. . . 4 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
173	172	2timesd 11868	. . 3 ⊢ (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174	171, 173	breqtrrd 5058	. 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 10786	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  supcsup 8888  ℂcc 10524  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  +∞cpnf 10661  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665   − cmin 10859   / cdiv 11286  ℕcn 11625  2c2 11680  ℝ⁺crp 12377  ...cfz 12885  ⌊cfl 13155  ↑cexp 13425  abscabs 14585   ⇝_𝑟 crli 14834  Σcsu 15034  eceu 15408  logclog 25146  γcem 25577

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-fi 8859  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-shft 14418  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-ef 15413  df-e 15414  df-sin 15415  df-cos 15416  df-tan 15417  df-pi 15418  df-dvds 15600  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-fbas 20088  df-fg 20089  df-cnfld 20092  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cld 21624  df-ntr 21625  df-cls 21626  df-nei 21703  df-lp 21741  df-perf 21742  df-cn 21832  df-cnp 21833  df-haus 21920  df-cmp 21992  df-tx 22167  df-hmeo 22360  df-fil 22451  df-fm 22543  df-flim 22544  df-flf 22545  df-xms 22927  df-ms 22928  df-tms 22929  df-cncf 23483  df-limc 24469  df-dv 24470  df-ulm 24972  df-log 25148  df-cxp 25149  df-atan 25453  df-em 25578

This theorem is referenced by:  mulog2sumlem2  26119