logdivsum - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > logdivsum		Structured version Visualization version GIF version

Theorem logdivsum 27577

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 13465	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2746	. . 3 ⊢ ℝ⁺ = (0(,)+∞)
3		nnuz 12921	. . 3 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12648	. . 3 ⊢ (⊤ → 1 ∈ ℤ)
5		ere 16125	. . . 4 ⊢ e ∈ ℝ
6	5	a1i 11	. . 3 ⊢ (⊤ → e ∈ ℝ)
7		0re 11263	. . . . . 6 ⊢ 0 ∈ ℝ
8		epos 16243	. . . . . 6 ⊢ 0 < e
9	7, 5, 8	ltleii 11384	. . . . 5 ⊢ 0 ≤ e
10	9	a1i 11	. . . 4 ⊢ (⊤ → 0 ≤ e)
11		1re 11261	. . . . 5 ⊢ 1 ∈ ℝ
12		addge02 11774	. . . . 5 ⊢ ((1 ∈ ℝ ∧ e ∈ ℝ) → (0 ≤ e ↔ 1 ≤ (e + 1)))
13	11, 5, 12	mp2an 692	. . . 4 ⊢ (0 ≤ e ↔ 1 ≤ (e + 1))
14	10, 13	sylib 218	. . 3 ⊢ (⊤ → 1 ≤ (e + 1))
15	7	a1i 11	. . 3 ⊢ (⊤ → 0 ∈ ℝ)
16		relogcl 26617	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
17	16	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
18	17	resqcld 14165	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦)↑2) ∈ ℝ)
19	18	rehalfcld 12513	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (((log‘𝑦)↑2) / 2) ∈ ℝ)
20		rerpdivcl 13065	. . . . 5 ⊢ (((log‘𝑦) ∈ ℝ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
21	16, 20	mpancom 688	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / 𝑦) ∈ ℝ)
22	21	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
23		nnrp 13046	. . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ⁺)
24	23, 22	sylan2 593	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℕ) → ((log‘𝑦) / 𝑦) ∈ ℝ)
25		reelprrecn 11247	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
26	25	a1i 11	. . . . 5 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
27		cnelprrecn 11248	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
28	27	a1i 11	. . . . 5 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
29	17	recnd 11289	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
30		ovexd 7466	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ V)
31		sqcl 14158	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
32	31	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑2) ∈ ℂ)
33	32	halfcld 12511	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((𝑥↑2) / 2) ∈ ℂ)
34		simpr 484	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
35		relogf1o 26608	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
36		f1of 6848	. . . . . . . . . 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 6977	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
39		fvres 6925	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
40	39	mpteq2ia 5245	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
41	38, 40	eqtrdi 2793	. . . . . . 7 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
42	41	oveq2d 7447	. . . . . 6 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
43		dvrelog 26679	. . . . . 6 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
44	42, 43	eqtr3di 2792	. . . . 5 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
45		ovexd 7466	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ V)
46		2nn 12339	. . . . . . . . 9 ⊢ 2 ∈ ℕ
47		dvexp 25991	. . . . . . . . 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 12392	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
50	49	oveq2i 7442	. . . . . . . . . . 11 ⊢ (𝑥↑(2 − 1)) = (𝑥↑1)
51		exp1 14108	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
52	51	adantl 481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑1) = 𝑥)
53	50, 52	eqtrid 2789	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑(2 − 1)) = 𝑥)
54	53	oveq2d 7447	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · (𝑥↑(2 − 1))) = (2 · 𝑥))
55	54	mpteq2dva 5242	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
56	48, 55	eqtrd 2777	. . . . . . 7 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
57		2cnd 12344	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
58		2ne0 12370	. . . . . . . 8 ⊢ 2 ≠ 0
59	58	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ≠ 0)
60	28, 32, 45, 56, 57, 59	dvmptdivc 26003	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)))
61		2cn 12341	. . . . . . . . 9 ⊢ 2 ∈ ℂ
62		divcan3 11948	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑥) / 2) = 𝑥)
63	61, 58, 62	mp3an23 1455	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((2 · 𝑥) / 2) = 𝑥)
64	63	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((2 · 𝑥) / 2) = 𝑥)
65	64	mpteq2dva 5242	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)) = (𝑥 ∈ ℂ ↦ 𝑥))
66	60, 65	eqtrd 2777	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ 𝑥))
67		oveq1 7438	. . . . . 6 ⊢ (𝑥 = (log‘𝑦) → (𝑥↑2) = ((log‘𝑦)↑2))
68	67	oveq1d 7446	. . . . 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 26010	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
71		rpcn 13045	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℂ)
72	71	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
73		rpne0 13051	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
74	73	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
75	29, 72, 74	divrecd 12046	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) = ((log‘𝑦) · (1 / 𝑦)))
76	75	mpteq2dva 5242	. . . 4 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
77	70, 76	eqtr4d 2780	. . 3 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)))
78		fveq2 6906	. . . 4 ⊢ (𝑦 = 𝑖 → (log‘𝑦) = (log‘𝑖))
79		id 22	. . . 4 ⊢ (𝑦 = 𝑖 → 𝑦 = 𝑖)
80	78, 79	oveq12d 7449	. . 3 ⊢ (𝑦 = 𝑖 → ((log‘𝑦) / 𝑦) = ((log‘𝑖) / 𝑖))
81		simp3r 1203	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ≤ 𝑖)
82		simp2l 1200	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ⁺)
83	82	rpred 13077	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ)
84		simp3l 1202	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑦)
85		simp2r 1201	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑖 ∈ ℝ⁺)
86	85	rpred 13077	. . . . 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 11418	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑖)
89		logdivle 26664	. . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ e ≤ 𝑦) ∧ (𝑖 ∈ ℝ ∧ e ≤ 𝑖)) → (𝑦 ≤ 𝑖 ↔ ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦)))
90	83, 84, 86, 88, 89	syl22anc 839	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → (𝑦 ≤ 𝑖 ↔ ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦)))
91	81, 90	mpbid 232	. . 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 26748	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦↑_𝑐1) = 𝑦)
94	93	oveq2d 7447	. . . . 5 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / (𝑦↑_𝑐1)) = ((log‘𝑦) / 𝑦))
95	94	mpteq2ia 5245	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / (𝑦↑_𝑐1))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦))
96		1rp 13038	. . . . 5 ⊢ 1 ∈ ℝ⁺
97		cxploglim 27021	. . . . 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 5179	. . 3 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) ⇝_𝑟 0)
100		fveq2 6906	. . . 4 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
101		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
102	100, 101	oveq12d 7449	. . 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 26074	. 2 ⊢ (⊤ → (𝐹:ℝ⁺⟶ℝ ∧ 𝐹 ∈ dom ⇝_𝑟 ∧ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))))
104	103	mptru 1547	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ↾ cres 5687 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 +∞cpnf 11292 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℝ⁺crp 13034 (,)cioo 13387 ...cfz 13547 ⌊cfl 13830 ↑cexp 14102 abscabs 15273 ⇝_𝑟 crli 15521 Σcsu 15722 eceu 16098 D cdv 25898 logclog 26596 ↑_𝑐ccxp 26597

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-e 16104 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599

This theorem is referenced by: mulog2sumlem1 27578 mulog2sum 27581 vmalogdivsum2 27582

W3C validator