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Theorem logdivsum 26414

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 13013	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2746	. . 3 ⊢ ℝ⁺ = (0(,)+∞)
3		nnuz 12477	. . 3 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12208	. . 3 ⊢ (⊤ → 1 ∈ ℤ)
5		ere 15650	. . . 4 ⊢ e ∈ ℝ
6	5	a1i 11	. . 3 ⊢ (⊤ → e ∈ ℝ)
7		0re 10835	. . . . . 6 ⊢ 0 ∈ ℝ
8		epos 15768	. . . . . 6 ⊢ 0 < e
9	7, 5, 8	ltleii 10955	. . . . 5 ⊢ 0 ≤ e
10	9	a1i 11	. . . 4 ⊢ (⊤ → 0 ≤ e)
11		1re 10833	. . . . 5 ⊢ 1 ∈ ℝ
12		addge02 11343	. . . . 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 221	. . 3 ⊢ (⊤ → 1 ≤ (e + 1))
15	7	a1i 11	. . 3 ⊢ (⊤ → 0 ∈ ℝ)
16		relogcl 25464	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
17	16	adantl 485	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
18	17	resqcld 13817	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦)↑2) ∈ ℝ)
19	18	rehalfcld 12077	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (((log‘𝑦)↑2) / 2) ∈ ℝ)
20		rerpdivcl 12616	. . . . 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 485	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
23		nnrp 12597	. . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ⁺)
24	23, 22	sylan2 596	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℕ) → ((log‘𝑦) / 𝑦) ∈ ℝ)
25		reelprrecn 10821	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
26	25	a1i 11	. . . . 5 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
27		cnelprrecn 10822	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
28	27	a1i 11	. . . . 5 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
29	17	recnd 10861	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
30		ovexd 7248	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ V)
31		sqcl 13690	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
32	31	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑2) ∈ ℂ)
33	32	halfcld 12075	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((𝑥↑2) / 2) ∈ ℂ)
34		simpr 488	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
35		relogf1o 25455	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
36		f1of 6661	. . . . . . . . . 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 6780	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
39		fvres 6736	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
40	39	mpteq2ia 5146	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
41	38, 40	eqtrdi 2794	. . . . . . 7 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
42	41	oveq2d 7229	. . . . . 6 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
43		dvrelog 25525	. . . . . 6 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
44	42, 43	eqtr3di 2793	. . . . 5 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
45		ovexd 7248	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ V)
46		2nn 11903	. . . . . . . . 9 ⊢ 2 ∈ ℕ
47		dvexp 24850	. . . . . . . . 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 11956	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
50	49	oveq2i 7224	. . . . . . . . . . 11 ⊢ (𝑥↑(2 − 1)) = (𝑥↑1)
51		exp1 13641	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
52	51	adantl 485	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑1) = 𝑥)
53	50, 52	syl5eq 2790	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑(2 − 1)) = 𝑥)
54	53	oveq2d 7229	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · (𝑥↑(2 − 1))) = (2 · 𝑥))
55	54	mpteq2dva 5150	. . . . . . . 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 11908	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
58		2ne0 11934	. . . . . . . 8 ⊢ 2 ≠ 0
59	58	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ≠ 0)
60	28, 32, 45, 56, 57, 59	dvmptdivc 24862	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)))
61		2cn 11905	. . . . . . . . 9 ⊢ 2 ∈ ℂ
62		divcan3 11516	. . . . . . . . 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 485	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((2 · 𝑥) / 2) = 𝑥)
65	64	mpteq2dva 5150	. . . . . 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 7220	. . . . . 6 ⊢ (𝑥 = (log‘𝑦) → (𝑥↑2) = ((log‘𝑦)↑2))
68	67	oveq1d 7228	. . . . 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 24869	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
71		rpcn 12596	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℂ)
72	71	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
73		rpne0 12602	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
74	73	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
75	29, 72, 74	divrecd 11611	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) = ((log‘𝑦) · (1 / 𝑦)))
76	75	mpteq2dva 5150	. . . 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 6717	. . . 4 ⊢ (𝑦 = 𝑖 → (log‘𝑦) = (log‘𝑖))
79		id 22	. . . 4 ⊢ (𝑦 = 𝑖 → 𝑦 = 𝑖)
80	78, 79	oveq12d 7231	. . 3 ⊢ (𝑦 = 𝑖 → ((log‘𝑦) / 𝑦) = ((log‘𝑖) / 𝑖))
81		simp3r 1204	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ≤ 𝑖)
82		simp2l 1201	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ⁺)
83	82	rpred 12628	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ)
84		simp3l 1203	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑦)
85		simp2r 1202	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑖 ∈ ℝ⁺)
86	85	rpred 12628	. . . . 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 10989	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑖)
89		logdivle 25510	. . . . 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 235	. . 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 25594	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦↑_𝑐1) = 𝑦)
94	93	oveq2d 7229	. . . . 5 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / (𝑦↑_𝑐1)) = ((log‘𝑦) / 𝑦))
95	94	mpteq2ia 5146	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / (𝑦↑_𝑐1))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦))
96		1rp 12590	. . . . 5 ⊢ 1 ∈ ℝ⁺
97		cxploglim 25860	. . . . 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 5089	. . 3 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) ⇝_𝑟 0)
100		fveq2 6717	. . . 4 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
101		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
102	100, 101	oveq12d 7231	. . 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 24930	. 2 ⊢ (⊤ → (𝐹:ℝ⁺⟶ℝ ∧ 𝐹 ∈ dom ⇝_𝑟 ∧ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))))
104	103	mptru 1550	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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2940 Vcvv 3408 {cpr 4543 class class class wbr 5053 ↦ cmpt 5135 dom cdm 5551 ↾ cres 5553 ⟶wf 6376 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 +∞cpnf 10864 ≤ cle 10868 − cmin 11062 / cdiv 11489 ℕcn 11830 2c2 11885 ℝ⁺crp 12586 (,)cioo 12935 ...cfz 13095 ⌊cfl 13365 ↑cexp 13635 abscabs 14797 ⇝_𝑟 crli 15046 Σcsu 15249 eceu 15624 D cdv 24760 logclog 25443 ↑_𝑐ccxp 25444

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629 df-e 15630 df-sin 15631 df-cos 15632 df-pi 15634 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-cmp 22284 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764 df-log 25445 df-cxp 25446

This theorem is referenced by: mulog2sumlem1 26415 mulog2sum 26418 vmalogdivsum2 26419

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