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Theorem logdivsum 25529

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 12460	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2774	. . 3 ⊢ ℝ⁺ = (0(,)+∞)
3		nnuz 11930	. . 3 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 11661	. . 3 ⊢ (⊤ → 1 ∈ ℤ)
5		ere 15117	. . . 4 ⊢ e ∈ ℝ
6	5	a1i 11	. . 3 ⊢ (⊤ → e ∈ ℝ)
7		0re 10299	. . . . . 6 ⊢ 0 ∈ ℝ
8		epos 15233	. . . . . 6 ⊢ 0 < e
9	7, 5, 8	ltleii 10419	. . . . 5 ⊢ 0 ≤ e
10	9	a1i 11	. . . 4 ⊢ (⊤ → 0 ≤ e)
11		1re 10297	. . . . 5 ⊢ 1 ∈ ℝ
12		addge02 10798	. . . . 5 ⊢ ((1 ∈ ℝ ∧ e ∈ ℝ) → (0 ≤ e ↔ 1 ≤ (e + 1)))
13	11, 5, 12	mp2an 683	. . . 4 ⊢ (0 ≤ e ↔ 1 ≤ (e + 1))
14	10, 13	sylib 209	. . 3 ⊢ (⊤ → 1 ≤ (e + 1))
15	7	a1i 11	. . 3 ⊢ (⊤ → 0 ∈ ℝ)
16		relogcl 24629	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
17	16	adantl 473	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
18	17	resqcld 13249	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦)↑2) ∈ ℝ)
19	18	rehalfcld 11530	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (((log‘𝑦)↑2) / 2) ∈ ℝ)
20		rerpdivcl 12066	. . . . 5 ⊢ (((log‘𝑦) ∈ ℝ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
21	16, 20	mpancom 679	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / 𝑦) ∈ ℝ)
22	21	adantl 473	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
23		nnrp 12048	. . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ⁺)
24	23, 22	sylan2 586	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℕ) → ((log‘𝑦) / 𝑦) ∈ ℝ)
25		reelprrecn 10285	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
26	25	a1i 11	. . . . 5 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
27		cnelprrecn 10286	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
28	27	a1i 11	. . . . 5 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
29	17	recnd 10326	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
30		ovexd 6880	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ V)
31		sqcl 13139	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
32	31	adantl 473	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑2) ∈ ℂ)
33	32	halfcld 11528	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((𝑥↑2) / 2) ∈ ℂ)
34		simpr 477	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
35		dvrelog 24690	. . . . . 6 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
36		relogf1o 24620	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
37		f1of 6324	. . . . . . . . . 10 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
38	36, 37	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
39	38	feqmptd 6442	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
40		fvres 6398	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
41	40	mpteq2ia 4901	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
42	39, 41	syl6eq 2815	. . . . . . 7 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
43	42	oveq2d 6862	. . . . . 6 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
44	35, 43	syl5reqr 2814	. . . . 5 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
45		ovexd 6880	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ V)
46		2nn 11350	. . . . . . . . 9 ⊢ 2 ∈ ℕ
47		dvexp 24023	. . . . . . . . 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 11410	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
50	49	oveq2i 6857	. . . . . . . . . . 11 ⊢ (𝑥↑(2 − 1)) = (𝑥↑1)
51		exp1 13080	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
52	51	adantl 473	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑1) = 𝑥)
53	50, 52	syl5eq 2811	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑(2 − 1)) = 𝑥)
54	53	oveq2d 6862	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · (𝑥↑(2 − 1))) = (2 · 𝑥))
55	54	mpteq2dva 4905	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
56	48, 55	eqtrd 2799	. . . . . . 7 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
57		2cn 11352	. . . . . . . 8 ⊢ 2 ∈ ℂ
58	57	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
59		2ne0 11388	. . . . . . . 8 ⊢ 2 ≠ 0
60	59	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ≠ 0)
61	28, 32, 45, 56, 58, 60	dvmptdivc 24035	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)))
62		divcan3 10970	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑥) / 2) = 𝑥)
63	57, 59, 62	mp3an23 1577	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((2 · 𝑥) / 2) = 𝑥)
64	63	adantl 473	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((2 · 𝑥) / 2) = 𝑥)
65	64	mpteq2dva 4905	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)) = (𝑥 ∈ ℂ ↦ 𝑥))
66	61, 65	eqtrd 2799	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ 𝑥))
67		oveq1 6853	. . . . . 6 ⊢ (𝑥 = (log‘𝑦) → (𝑥↑2) = ((log‘𝑦)↑2))
68	67	oveq1d 6861	. . . . 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 24042	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
71		rpcn 12047	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℂ)
72	71	adantl 473	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
73		rpne0 12053	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
74	73	adantl 473	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
75	29, 72, 74	divrecd 11063	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) = ((log‘𝑦) · (1 / 𝑦)))
76	75	mpteq2dva 4905	. . . 4 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
77	70, 76	eqtr4d 2802	. . 3 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)))
78		fveq2 6379	. . . 4 ⊢ (𝑦 = 𝑖 → (log‘𝑦) = (log‘𝑖))
79		id 22	. . . 4 ⊢ (𝑦 = 𝑖 → 𝑦 = 𝑖)
80	78, 79	oveq12d 6864	. . 3 ⊢ (𝑦 = 𝑖 → ((log‘𝑦) / 𝑦) = ((log‘𝑖) / 𝑖))
81		simp3r 1259	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ≤ 𝑖)
82		simp2l 1256	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ⁺)
83	82	rpred 12077	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ)
84		simp3l 1258	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑦)
85		simp2r 1257	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑖 ∈ ℝ⁺)
86	85	rpred 12077	. . . . 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 10453	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑖)
89		logdivle 24675	. . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ e ≤ 𝑦) ∧ (𝑖 ∈ ℝ ∧ e ≤ 𝑖)) → (𝑦 ≤ 𝑖 ↔ ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦)))
90	83, 84, 86, 88, 89	syl22anc 867	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → (𝑦 ≤ 𝑖 ↔ ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦)))
91	81, 90	mpbid 223	. . 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 24759	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦↑_𝑐1) = 𝑦)
94	93	oveq2d 6862	. . . . 5 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / (𝑦↑_𝑐1)) = ((log‘𝑦) / 𝑦))
95	94	mpteq2ia 4901	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / (𝑦↑_𝑐1))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦))
96		1rp 12039	. . . . 5 ⊢ 1 ∈ ℝ⁺
97		cxploglim 25011	. . . . 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	syl5eqbrr 4847	. . 3 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) ⇝_𝑟 0)
100		fveq2 6379	. . . 4 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
101		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
102	100, 101	oveq12d 6864	. . 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 24103	. 2 ⊢ (⊤ → (𝐹:ℝ⁺⟶ℝ ∧ 𝐹 ∈ dom ⇝_𝑟 ∧ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))))
104	103	mptru 1660	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 197 ∧ wa 384 ∧ w3a 1107 = wceq 1652 ⊤wtru 1653 ∈ wcel 2155 ≠ wne 2937 Vcvv 3350 {cpr 4338 class class class wbr 4811 ↦ cmpt 4890 dom cdm 5279 ↾ cres 5281 ⟶wf 6066 –1-1-onto→wf1o 6069 ‘cfv 6070 (class class class)co 6846 ℂcc 10191 ℝcr 10192 0cc0 10193 1c1 10194 + caddc 10196 · cmul 10198 +∞cpnf 10329 ≤ cle 10333 − cmin 10525 / cdiv 10943 ℕcn 11279 2c2 11332 ℝ⁺crp 12035 (,)cioo 12384 ...cfz 12540 ⌊cfl 12806 ↑cexp 13074 abscabs 14275 ⇝_𝑟 crli 14517 Σcsu 14717 eceu 15091 D cdv 23934 logclog 24608 ↑_𝑐ccxp 24609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2070 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-rep 4932 ax-sep 4943 ax-nul 4951 ax-pow 5003 ax-pr 5064 ax-un 7151 ax-inf2 8757 ax-cnex 10249 ax-resscn 10250 ax-1cn 10251 ax-icn 10252 ax-addcl 10253 ax-addrcl 10254 ax-mulcl 10255 ax-mulrcl 10256 ax-mulcom 10257 ax-addass 10258 ax-mulass 10259 ax-distr 10260 ax-i2m1 10261 ax-1ne0 10262 ax-1rid 10263 ax-rnegex 10264 ax-rrecex 10265 ax-cnre 10266 ax-pre-lttri 10267 ax-pre-lttrn 10268 ax-pre-ltadd 10269 ax-pre-mulgt0 10270 ax-pre-sup 10271 ax-addf 10272 ax-mulf 10273

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2063 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rmo 3063 df-rab 3064 df-v 3352 df-sbc 3599 df-csb 3694 df-dif 3737 df-un 3739 df-in 3741 df-ss 3748 df-pss 3750 df-nul 4082 df-if 4246 df-pw 4319 df-sn 4337 df-pr 4339 df-tp 4341 df-op 4343 df-uni 4597 df-int 4636 df-iun 4680 df-iin 4681 df-br 4812 df-opab 4874 df-mpt 4891 df-tr 4914 df-id 5187 df-eprel 5192 df-po 5200 df-so 5201 df-fr 5238 df-se 5239 df-we 5240 df-xp 5285 df-rel 5286 df-cnv 5287 df-co 5288 df-dm 5289 df-rn 5290 df-res 5291 df-ima 5292 df-pred 5867 df-ord 5913 df-on 5914 df-lim 5915 df-suc 5916 df-iota 6033 df-fun 6072 df-fn 6073 df-f 6074 df-f1 6075 df-fo 6076 df-f1o 6077 df-fv 6078 df-isom 6079 df-riota 6807 df-ov 6849 df-oprab 6850 df-mpt2 6851 df-of 7099 df-om 7268 df-1st 7370 df-2nd 7371 df-supp 7502 df-wrecs 7614 df-recs 7676 df-rdg 7714 df-1o 7768 df-2o 7769 df-oadd 7772 df-er 7951 df-map 8066 df-pm 8067 df-ixp 8118 df-en 8165 df-dom 8166 df-sdom 8167 df-fin 8168 df-fsupp 8487 df-fi 8528 df-sup 8559 df-inf 8560 df-oi 8626 df-card 9020 df-cda 9247 df-pnf 10334 df-mnf 10335 df-xr 10336 df-ltxr 10337 df-le 10338 df-sub 10527 df-neg 10528 df-div 10944 df-nn 11280 df-2 11340 df-3 11341 df-4 11342 df-5 11343 df-6 11344 df-7 11345 df-8 11346 df-9 11347 df-n0 11544 df-z 11630 df-dec 11747 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12153 df-xadd 12154 df-xmul 12155 df-ioo 12388 df-ioc 12389 df-ico 12390 df-icc 12391 df-fz 12541 df-fzo 12681 df-fl 12808 df-mod 12884 df-seq 13016 df-exp 13075 df-fac 13272 df-bc 13301 df-hash 13329 df-shft 14108 df-cj 14140 df-re 14141 df-im 14142 df-sqrt 14276 df-abs 14277 df-limsup 14503 df-clim 14520 df-rlim 14521 df-sum 14718 df-ef 15096 df-e 15097 df-sin 15098 df-cos 15099 df-pi 15101 df-struct 16148 df-ndx 16149 df-slot 16150 df-base 16152 df-sets 16153 df-ress 16154 df-plusg 16243 df-mulr 16244 df-starv 16245 df-sca 16246 df-vsca 16247 df-ip 16248 df-tset 16249 df-ple 16250 df-ds 16252 df-unif 16253 df-hom 16254 df-cco 16255 df-rest 16365 df-topn 16366 df-0g 16384 df-gsum 16385 df-topgen 16386 df-pt 16387 df-prds 16390 df-xrs 16444 df-qtop 16449 df-imas 16450 df-xps 16452 df-mre 16528 df-mrc 16529 df-acs 16531 df-mgm 17524 df-sgrp 17566 df-mnd 17577 df-submnd 17618 df-mulg 17824 df-cntz 18029 df-cmn 18477 df-psmet 20027 df-xmet 20028 df-met 20029 df-bl 20030 df-mopn 20031 df-fbas 20032 df-fg 20033 df-cnfld 20036 df-top 20994 df-topon 21011 df-topsp 21033 df-bases 21046 df-cld 21119 df-ntr 21120 df-cls 21121 df-nei 21198 df-lp 21236 df-perf 21237 df-cn 21327 df-cnp 21328 df-haus 21415 df-cmp 21486 df-tx 21661 df-hmeo 21854 df-fil 21945 df-fm 22037 df-flim 22038 df-flf 22039 df-xms 22420 df-ms 22421 df-tms 22422 df-cncf 22976 df-limc 23937 df-dv 23938 df-log 24610 df-cxp 24611

This theorem is referenced by: mulog2sumlem1 25530 mulog2sum 25533 vmalogdivsum2 25534

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