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Theorem logdivsum 27559

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 13450	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2735	. . 3 ⊢ ℝ⁺ = (0(,)+∞)
3		nnuz 12911	. . 3 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12639	. . 3 ⊢ (⊤ → 1 ∈ ℤ)
5		ere 16086	. . . 4 ⊢ e ∈ ℝ
6	5	a1i 11	. . 3 ⊢ (⊤ → e ∈ ℝ)
7		0re 11257	. . . . . 6 ⊢ 0 ∈ ℝ
8		epos 16204	. . . . . 6 ⊢ 0 < e
9	7, 5, 8	ltleii 11378	. . . . 5 ⊢ 0 ≤ e
10	9	a1i 11	. . . 4 ⊢ (⊤ → 0 ≤ e)
11		1re 11255	. . . . 5 ⊢ 1 ∈ ℝ
12		addge02 11766	. . . . 5 ⊢ ((1 ∈ ℝ ∧ e ∈ ℝ) → (0 ≤ e ↔ 1 ≤ (e + 1)))
13	11, 5, 12	mp2an 690	. . . 4 ⊢ (0 ≤ e ↔ 1 ≤ (e + 1))
14	10, 13	sylib 217	. . 3 ⊢ (⊤ → 1 ≤ (e + 1))
15	7	a1i 11	. . 3 ⊢ (⊤ → 0 ∈ ℝ)
16		relogcl 26599	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
17	16	adantl 480	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
18	17	resqcld 14138	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦)↑2) ∈ ℝ)
19	18	rehalfcld 12505	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (((log‘𝑦)↑2) / 2) ∈ ℝ)
20		rerpdivcl 13052	. . . . 5 ⊢ (((log‘𝑦) ∈ ℝ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
21	16, 20	mpancom 686	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / 𝑦) ∈ ℝ)
22	21	adantl 480	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) ∈ ℝ)
23		nnrp 13033	. . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ⁺)
24	23, 22	sylan2 591	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℕ) → ((log‘𝑦) / 𝑦) ∈ ℝ)
25		reelprrecn 11241	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
26	25	a1i 11	. . . . 5 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
27		cnelprrecn 11242	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
28	27	a1i 11	. . . . 5 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
29	17	recnd 11283	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
30		ovexd 7451	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ V)
31		sqcl 14131	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
32	31	adantl 480	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑2) ∈ ℂ)
33	32	halfcld 12503	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((𝑥↑2) / 2) ∈ ℂ)
34		simpr 483	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
35		relogf1o 26590	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
36		f1of 6835	. . . . . . . . . 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 6963	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
39		fvres 6912	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
40	39	mpteq2ia 5248	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
41	38, 40	eqtrdi 2782	. . . . . . 7 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
42	41	oveq2d 7432	. . . . . 6 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
43		dvrelog 26661	. . . . . 6 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
44	42, 43	eqtr3di 2781	. . . . 5 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
45		ovexd 7451	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ V)
46		2nn 12331	. . . . . . . . 9 ⊢ 2 ∈ ℕ
47		dvexp 25973	. . . . . . . . 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 12384	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
50	49	oveq2i 7427	. . . . . . . . . . 11 ⊢ (𝑥↑(2 − 1)) = (𝑥↑1)
51		exp1 14081	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
52	51	adantl 480	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑1) = 𝑥)
53	50, 52	eqtrid 2778	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑(2 − 1)) = 𝑥)
54	53	oveq2d 7432	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · (𝑥↑(2 − 1))) = (2 · 𝑥))
55	54	mpteq2dva 5245	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
56	48, 55	eqtrd 2766	. . . . . . 7 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
57		2cnd 12336	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
58		2ne0 12362	. . . . . . . 8 ⊢ 2 ≠ 0
59	58	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ≠ 0)
60	28, 32, 45, 56, 57, 59	dvmptdivc 25985	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)))
61		2cn 12333	. . . . . . . . 9 ⊢ 2 ∈ ℂ
62		divcan3 11940	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑥) / 2) = 𝑥)
63	61, 58, 62	mp3an23 1450	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((2 · 𝑥) / 2) = 𝑥)
64	63	adantl 480	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((2 · 𝑥) / 2) = 𝑥)
65	64	mpteq2dva 5245	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((2 · 𝑥) / 2)) = (𝑥 ∈ ℂ ↦ 𝑥))
66	60, 65	eqtrd 2766	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 2))) = (𝑥 ∈ ℂ ↦ 𝑥))
67		oveq1 7423	. . . . . 6 ⊢ (𝑥 = (log‘𝑦) → (𝑥↑2) = ((log‘𝑦)↑2))
68	67	oveq1d 7431	. . . . 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 25992	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
71		rpcn 13032	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℂ)
72	71	adantl 480	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
73		rpne0 13038	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
74	73	adantl 480	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
75	29, 72, 74	divrecd 12038	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ⁺) → ((log‘𝑦) / 𝑦) = ((log‘𝑦) · (1 / 𝑦)))
76	75	mpteq2dva 5245	. . . 4 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) · (1 / 𝑦))))
77	70, 76	eqtr4d 2769	. . 3 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (((log‘𝑦)↑2) / 2))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)))
78		fveq2 6893	. . . 4 ⊢ (𝑦 = 𝑖 → (log‘𝑦) = (log‘𝑖))
79		id 22	. . . 4 ⊢ (𝑦 = 𝑖 → 𝑦 = 𝑖)
80	78, 79	oveq12d 7434	. . 3 ⊢ (𝑦 = 𝑖 → ((log‘𝑦) / 𝑦) = ((log‘𝑖) / 𝑖))
81		simp3r 1199	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ≤ 𝑖)
82		simp2l 1196	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ⁺)
83	82	rpred 13064	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑦 ∈ ℝ)
84		simp3l 1198	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑦)
85		simp2r 1197	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → 𝑖 ∈ ℝ⁺)
86	85	rpred 13064	. . . . 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 11412	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → e ≤ 𝑖)
89		logdivle 26646	. . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ e ≤ 𝑦) ∧ (𝑖 ∈ ℝ ∧ e ≤ 𝑖)) → (𝑦 ≤ 𝑖 ↔ ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦)))
90	83, 84, 86, 88, 89	syl22anc 837	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑖 ∈ ℝ⁺) ∧ (e ≤ 𝑦 ∧ 𝑦 ≤ 𝑖)) → (𝑦 ≤ 𝑖 ↔ ((log‘𝑖) / 𝑖) ≤ ((log‘𝑦) / 𝑦)))
91	81, 90	mpbid 231	. . 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 26730	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦↑_𝑐1) = 𝑦)
94	93	oveq2d 7432	. . . . 5 ⊢ (𝑦 ∈ ℝ⁺ → ((log‘𝑦) / (𝑦↑_𝑐1)) = ((log‘𝑦) / 𝑦))
95	94	mpteq2ia 5248	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / (𝑦↑_𝑐1))) = (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦))
96		1rp 13026	. . . . 5 ⊢ 1 ∈ ℝ⁺
97		cxploglim 27003	. . . . 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 5181	. . 3 ⊢ (⊤ → (𝑦 ∈ ℝ⁺ ↦ ((log‘𝑦) / 𝑦)) ⇝_𝑟 0)
100		fveq2 6893	. . . 4 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
101		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
102	100, 101	oveq12d 7434	. . 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 26056	. 2 ⊢ (⊤ → (𝐹:ℝ⁺⟶ℝ ∧ 𝐹 ∈ dom ⇝_𝑟 ∧ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))))
104	103	mptru 1541	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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 {cpr 4625 class class class wbr 5145 ↦ cmpt 5228 dom cdm 5674 ↾ cres 5676 ⟶wf 6542 –1-1-onto→wf1o 6545 ‘cfv 6546 (class class class)co 7416 ℂcc 11147 ℝcr 11148 0cc0 11149 1c1 11150 + caddc 11152 · cmul 11154 +∞cpnf 11286 ≤ cle 11290 − cmin 11485 / cdiv 11912 ℕcn 12258 2c2 12313 ℝ⁺crp 13022 (,)cioo 13372 ...cfz 13532 ⌊cfl 13804 ↑cexp 14075 abscabs 15234 ⇝_𝑟 crli 15482 Σcsu 15685 eceu 16059 D cdv 25880 logclog 26578 ↑_𝑐ccxp 26579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13533 df-fzo 13676 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-fac 14286 df-bc 14315 df-hash 14343 df-shft 15067 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-limsup 15468 df-clim 15485 df-rlim 15486 df-sum 15686 df-ef 16064 df-e 16065 df-sin 16066 df-cos 16067 df-pi 16069 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19058 df-cntz 19307 df-cmn 19776 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-fbas 21336 df-fg 21337 df-cnfld 21340 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-nei 23090 df-lp 23128 df-perf 23129 df-cn 23219 df-cnp 23220 df-haus 23307 df-cmp 23379 df-tx 23554 df-hmeo 23747 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-xms 24314 df-ms 24315 df-tms 24316 df-cncf 24886 df-limc 25883 df-dv 25884 df-log 26580 df-cxp 26581

This theorem is referenced by: mulog2sumlem1 27560 mulog2sum 27563 vmalogdivsum2 27564

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