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Theorem logdivsum 27563

Description: Asymptotic analysis of Σ𝑛 ≤ 𝑥, log𝑛 / 𝑛 = (log𝑥)↑2 / 2 + 𝐿 + 𝑂(log𝑥 / 𝑥). (Contributed by Mario Carneiro, 18-May-2016.)

**Hypothesis**
Ref	Expression
logdivsum.1	⊢ 𝐹 = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))

**Assertion**
Ref	Expression
logdivsum	⊢ (𝐹:ℝ⁺⟶ℝ ∧ 𝐹 ∈ dom ⇝_𝑟 ∧ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))

Distinct variable group: 𝑦,𝑖,𝐴 Allowed substitution hints: 𝐹(𝑦,𝑖) 𝐿(𝑦,𝑖)

Proof of Theorem logdivsum Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioorp 13415	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2761	. . 3 ⊢ ℝ⁺ = (0(,)+∞)
3		nnuz 12864	. . 3 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12588	. . 3 ⊢ (⊤ → 1 ∈ ℤ)
5		ere 16091	. . . 4 ⊢ e ∈ ℝ
6	5	a1i 11	. . 3 ⊢ (⊤ → e ∈ ℝ)
7		0re 11169	. . . . . 6 ⊢ 0 ∈ ℝ
8		epos 16211	. . . . . 6 ⊢ 0 < e
9	7, 5, 8	ltleii 11292	. . . . 5 ⊢ 0 ≤ e
10	9	a1i 11	. . . 4 ⊢ (⊤ → 0 ≤ e)
11		1re 11167	. . . . 5 ⊢ 1 ∈ ℝ
12		addge02 11684	. . . . 5 ⊢ ((1 ∈ ℝ ∧ e ∈ ℝ) → (0 ≤ e ↔ 1 ≤ (e + 1)))
13	11, 5, 12	mp2an 700	. . . 4 ⊢ (0 ≤ e ↔ 1 ≤ (e + 1))
14	10, 13	sylib 220	. . 3 ⊢ (⊤ → 1 ≤ (e + 1))
15	7	a1i 11	. . 3 ⊢ (⊤ → 0 ∈ ℝ)
16		relogcl 26606	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
17	16	adantl 484	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
18	17	resqcld 14124	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦)↑2) ∈ ℝ)
19	18	rehalfcld 12454	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (((log‘𝑦)↑2) / 2) ∈ ℝ)
20		rerpdivcl 13011	. . . . 5 ⊢ (((log‘𝑦) ∈ ℝ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
21	16, 20	mpancom 696	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / 𝑦) ∈ ℝ)
22	21	adantl 484	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
23		nnrp 12991	. . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ⁺)
24	23, 22	sylan2 601	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℕ) → ((log‘𝑦) / 𝑦) ∈ ℝ)
25		reelprrecn 11151	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
26	25	a1i 11	. . . . 5 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
27		cnelprrecn 11152	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
28	27	a1i 11	. . . . 5 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
29	17	recnd 11196	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
30		ovexd 7416	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ V)
31		sqcl 14117	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
32	31	adantl 484	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑2) ∈ ℂ)
33	32	halfcld 12452	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((𝑥↑2) / 2) ∈ ℂ)
34		simpr 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
35		relogf1o 26597	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
36		f1of 6791	. . . . . . . . . 10 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
37	35, 36	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
38	37	feqmptd 6920	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
39		fvres 6871	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
40	39	mpteq2ia 5185	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
41	38, 40	eqtrdi 2803	. . . . . . 7 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
42	41	oveq2d 7397	. . . . . 6 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
43		dvrelog 26668	. . . . . 6 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
44	42, 43	eqtr3di 2802	. . . . 5 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
45		ovexd 7416	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ V)
46		2nn 12277	. . . . . . . . 9 ⊢ 2 ∈ ℕ
47		dvexp 25984	. . . . . . . . 9 ⊢ (2 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))))
48	46, 47	mp1i 13	. . . . . . . 8 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))))
49		2m1e1 12328	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
50	49	oveq2i 7392	. . . . . . . . . . 11 ⊢ (𝑥↑(2 − 1)) = (𝑥↑1)
51		exp1 14066	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
52	51	adantl 484	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑1) = 𝑥)
53	50, 52	eqtrid 2799	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑(2 − 1)) = 𝑥)
54	53	oveq2d 7397	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · (𝑥↑(2 − 1))) = (2 · 𝑥))
55	54	mpteq2dva 5183	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
56	48, 55	eqtrd 2787	. . . . . . 7 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
57		2cnd 12282	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
58		2ne0 12310	. . . . . . . 8 ⊢ 2 ≠ 0
59	58	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ≠ 0)
60	28, 32, 45, 56, 57, 59	dvmptdivc 25996	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)))
61		2cn 12279	. . . . . . . . 9 ⊢ 2 ∈ ℂ
62		divcan3 11857	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑥) / 2) = 𝑥)
63	61, 58, 62	mp3an23 1464	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((2 · 𝑥) / 2) = 𝑥)
64	63	adantl 484	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((2 · 𝑥) / 2) = 𝑥)
65	64	mpteq2dva 5183	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)) = (𝑥 ∈ ℂ ↦ 𝑥))
66	60, 65	eqtrd 2787	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ 𝑥))
67		oveq1 7388	. . . . . 6 ⊢ (𝑥 = (log‘𝑦) → (𝑥↑2) = ((log‘𝑦)↑2))
68	67	oveq1d 7396	. . . . 5 ⊢ (𝑥 = (log‘𝑦) → ((𝑥↑2) / 2) = (((log‘𝑦)↑2) / 2))
69		id 22	. . . . 5 ⊢ (𝑥 = (log‘𝑦) → 𝑥 = (log‘𝑦))
70	26, 28, 29, 30, 33, 34, 44, 66, 68, 69	dvmptco 26003	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
71		rpcn 12990	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℂ)
72	71	adantl 484	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
73		rpne0 12996	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
74	73	adantl 484	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
75	29, 72, 74	divrecd 11956	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) = ((log‘𝑦) · (1 / 𝑦)))
76	75	mpteq2dva 5183	. . . 4 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
77	70, 76	eqtr4d 2790	. . 3 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)))
78		fveq2 6852	. . . 4 ⊢ (𝑦 = 𝑖 → (log‘𝑦) = (log‘𝑖))
79		id 22	. . . 4 ⊢ (𝑦 = 𝑖 → 𝑦 = 𝑖)
80	78, 79	oveq12d 7399	. . 3 ⊢ (𝑦 = 𝑖 → ((log‘𝑦) / 𝑦) = ((log‘𝑖) / 𝑖))
81		simp3r 1212	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ≤ 𝑖)
82		simp2l 1209	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ⁺)
83	82	rpred 13023	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ)
84		simp3l 1211	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑦)
85		simp2r 1210	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑖 ∈ ℝ⁺)
86	85	rpred 13023	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑖 ∈ ℝ)
87	5	a1i 11	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ∈ ℝ)
88	87, 83, 86, 84, 81	letrd 11326	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑖)
89		logdivle 26653	. . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ e ≤ 𝑦) ∧ (𝑖 ∈ ℝ ∧ e ≤ 𝑖)) → (𝑦 ≤ 𝑖 ↔ ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦)))
90	83, 84, 86, 88, 89	syl22anc 847	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → (𝑦 ≤ 𝑖 ↔ ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦)))
91	81, 90	mpbid 234	. . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦))
92		logdivsum.1	. . 3 ⊢ 𝐹 = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))
93	71	cxp1d 26737	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦↑_𝑐1) = 𝑦)
94	93	oveq2d 7397	. . . . 5 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / (𝑦↑_𝑐1)) = ((log‘𝑦) / 𝑦))
95	94	mpteq2ia 5185	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / (𝑦↑_𝑐1))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦))
96		1rp 12983	. . . . 5 ⊢ 1 ∈ ℝ⁺
97		cxploglim 27008	. . . . 5 ⊢ (1 ∈ ℝ⁺ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / (𝑦↑_𝑐1))) ⇝_𝑟 0)
98	96, 97	mp1i 13	. . . 4 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / (𝑦↑_𝑐1))) ⇝_𝑟 0)
99	95, 98	eqbrtrrid 5126	. . 3 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) ⇝_𝑟 0)
100		fveq2 6852	. . . 4 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
101		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
102	100, 101	oveq12d 7399	. . 3 ⊢ (𝑦 = 𝐴 → ((log‘𝑦) / 𝑦) = ((log‘𝐴) / 𝐴))
103	2, 3, 4, 6, 14, 15, 19, 22, 24, 77, 80, 91, 92, 99, 102	dvfsumrlim3 26064	. 2 ⊢ (⊤ → (𝐹:ℝ⁺⟶ℝ ∧ 𝐹 ∈ dom ⇝_𝑟 ∧ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))))
104	103	mptru 1557	1 ⊢ (𝐹:ℝ⁺⟶ℝ ∧ 𝐹 ∈ dom ⇝_𝑟 ∧ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1095 = wceq 1550 ⊤wtru 1551 ∈ wcel 2132 ≠ wne 2947 Vcvv 3444 {cpr 4574 class class class wbr 5090 ↦ cmpt 5171 dom cdm 5636 ↾ cres 5638 ⟶wf 6502 –1-1-onto→wf1o 6505 ‘cfv 6506 (class class class)co 7381 ℂcc 11057 ℝcr 11058 0cc0 11059 1c1 11060 + caddc 11062 · cmul 11064 +∞cpnf 11199 ≤ cle 11203 − cmin 11400 / cdiv 11830 ℕcn 12196 2c2 12258 ℝ⁺crp 12979 (,)cioo 13335 ...cfz 13498 ⌊cfl 13786 ↑cexp 14060 abscabs 15233 ⇝_𝑟 crli 15484 Σcsu 15685 eceu 16064 D cdv 25894 logclog 26585 ↑_𝑐ccxp 26586

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-inf2 9582 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-er 8662 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9444 df-card 9883 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-ioo 13339 df-ioc 13340 df-ico 13341 df-icc 13342 df-fz 13499 df-fzo 13646 df-fl 13788 df-mod 13866 df-seq 14001 df-exp 14061 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15066 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-limsup 15470 df-clim 15487 df-rlim 15488 df-sum 15686 df-ef 16069 df-e 16070 df-sin 16071 df-cos 16072 df-pi 16074 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17504 df-qtop 17509 df-imas 17510 df-xps 17512 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-submnd 18790 df-mulg 19082 df-cntz 19329 df-cmn 19794 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-fbas 21390 df-fg 21391 df-cnfld 21394 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24349 df-ms 24350 df-tms 24351 df-cncf 24909 df-limc 25897 df-dv 25898 df-log 26587 df-cxp 26588

This theorem is referenced by: mulog2sumlem1 27564 mulog2sum 27567 vmalogdivsum2 27568

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