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Theorem mulog2sumlem1 27270

Description: Asymptotic formula for Σ𝑛 ≤ 𝑥, log(𝑥 / 𝑛) / 𝑛 = (1 / 2)log↑2(𝑥) + γ · log𝑥 − 𝐿 + 𝑂(log𝑥 / 𝑥), with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)

**Hypotheses**
Ref	Expression
logdivsum.1	⊢ 𝐹 = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))
mulog2sumlem.1	⊢ (𝜑 → 𝐹 ⇝_𝑟 𝐿)
mulog2sumlem1.2	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
mulog2sumlem1.3	⊢ (𝜑 → e ≤ 𝐴)

**Assertion**
Ref	Expression
mulog2sumlem1	⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ≤ (2 · ((log‘𝐴) / 𝐴)))

Distinct variable groups: 𝑖,𝑚,𝑦,𝐴 𝜑,𝑚 Allowed substitution hints: 𝜑(𝑦,𝑖) 𝐹(𝑦,𝑖,𝑚) 𝐿(𝑦,𝑖,𝑚)

Proof of Theorem mulog2sumlem1 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13943	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		mulog2sumlem1.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
3		elfznn 13535	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
4	3	nnrpd 13019	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℝ⁺)
5		rpdivcl 13004	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝐴 / 𝑚) ∈ ℝ⁺)
6	2, 4, 5	syl2an 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑚) ∈ ℝ⁺)
7	6	relogcld 26364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) ∈ ℝ)
8	3	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
9	7, 8	nndivred 12271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
10	1, 9	fsumrecl 15685	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) ∈ ℝ)
11	2	relogcld 26364	. . . . . . . 8 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
12	11	resqcld 14095	. . . . . . 7 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℝ)
13	12	rehalfcld 12464	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℝ)
14		emre 26743	. . . . . . . 8 ⊢ γ ∈ ℝ
15		remulcl 11198	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (γ · (log‘𝐴)) ∈ ℝ)
16	14, 11, 15	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℝ)
17		rpsup 13836	. . . . . . . . 9 ⊢ sup(ℝ⁺, ℝ^*, < ) = +∞
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → sup(ℝ⁺, ℝ^*, < ) = +∞)
19		logdivsum.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)))
20	19	logdivsum 27269	. . . . . . . . . . . 12 ⊢ (𝐹:ℝ⁺⟶ℝ ∧ 𝐹 ∈ dom ⇝_𝑟 ∧ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴)))
21	20	simp1i 1138	. . . . . . . . . . 11 ⊢ 𝐹:ℝ⁺⟶ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ⁺⟶ℝ)
23	22	feqmptd 6961	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ⁺ ↦ (𝐹‘𝑥)))
24		mulog2sumlem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⇝_𝑟 𝐿)
25	23, 24	eqbrtrrd 5173	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝐹‘𝑥)) ⇝_𝑟 𝐿)
26	21	ffvelcdmi 7086	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝐹‘𝑥) ∈ ℝ)
27	26	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐹‘𝑥) ∈ ℝ)
28	18, 25, 27	rlimrecl 15529	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ)
29	16, 28	resubcld 11647	. . . . . 6 ⊢ (𝜑 → ((γ · (log‘𝐴)) − 𝐿) ∈ ℝ)
30	13, 29	readdcld 11248	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) ∈ ℝ)
31	10, 30	resubcld 11647	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℝ)
32	31	recnd 11247	. . 3 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) ∈ ℂ)
33	32	abscld 15388	. 2 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ∈ ℝ)
34		rerpdivcl 13009	. . . . . . . 8 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝑚 ∈ ℝ⁺) → ((log‘𝐴) / 𝑚) ∈ ℝ)
35	11, 4, 34	syl2an 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℝ)
36	35	recnd 11247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) ∈ ℂ)
37	1, 36	fsumcl 15684	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) ∈ ℂ)
38	11	recnd 11247	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
39		readdcl 11196	. . . . . . . 8 ⊢ (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) + γ) ∈ ℝ)
40	11, 14, 39	sylancl 585	. . . . . . 7 ⊢ (𝜑 → ((log‘𝐴) + γ) ∈ ℝ)
41	40	recnd 11247	. . . . . 6 ⊢ (𝜑 → ((log‘𝐴) + γ) ∈ ℂ)
42	38, 41	mulcld 11239	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) ∈ ℂ)
43	37, 42	subcld 11576	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) ∈ ℂ)
44	43	abscld 15388	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ∈ ℝ)
45	8	nnrpd 13019	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℝ⁺)
46	45	relogcld 26364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℝ)
47	46, 8	nndivred 12271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
48	47	recnd 11247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
49	1, 48	fsumcl 15684	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) ∈ ℂ)
50	13	recnd 11247	. . . . . 6 ⊢ (𝜑 → (((log‘𝐴)↑2) / 2) ∈ ℂ)
51	28	recnd 11247	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℂ)
52	50, 51	addcld 11238	. . . . 5 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + 𝐿) ∈ ℂ)
53	49, 52	subcld 11576	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)) ∈ ℂ)
54	53	abscld 15388	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ∈ ℝ)
55	44, 54	readdcld 11248	. 2 ⊢ (𝜑 → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))) ∈ ℝ)
56		2re 12291	. . 3 ⊢ 2 ∈ ℝ
57	11, 2	rerpdivcld 13052	. . 3 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℝ)
58		remulcl 11198	. . 3 ⊢ ((2 ∈ ℝ ∧ ((log‘𝐴) / 𝐴) ∈ ℝ) → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
59	56, 57, 58	sylancr 586	. 2 ⊢ (𝜑 → (2 · ((log‘𝐴) / 𝐴)) ∈ ℝ)
60		relogdiv 26334	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
61	2, 4, 60	syl2an 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑚)) = ((log‘𝐴) − (log‘𝑚)))
62	61	oveq1d 7427	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) − (log‘𝑚)) / 𝑚))
63	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝐴) ∈ ℂ)
64	46	recnd 11247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (log‘𝑚) ∈ ℂ)
65	45	rpcnne0d 13030	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
66		divsubdir 11913	. . . . . . . . . 10 ⊢ (((log‘𝐴) ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
67	63, 64, 65, 66	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (((log‘𝐴) − (log‘𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
68	62, 67	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘(𝐴 / 𝑚)) / 𝑚) = (((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
69	68	sumeq2dv 15654	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)))
70	1, 36, 48	fsumsub 15739	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(((log‘𝐴) / 𝑚) − ((log‘𝑚) / 𝑚)) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
71	69, 70	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)))
72		remulcl 11198	. . . . . . . . . . . . 13 ⊢ (((log‘𝐴) ∈ ℝ ∧ γ ∈ ℝ) → ((log‘𝐴) · γ) ∈ ℝ)
73	11, 14, 72	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) · γ) ∈ ℝ)
74	13, 73	readdcld 11248	. . . . . . . . . . 11 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℝ)
75	74	recnd 11247	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) ∈ ℂ)
76	75, 50	pncand 11577	. . . . . . . . 9 ⊢ (𝜑 → ((((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)) − (((log‘𝐴)↑2) / 2)) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
77	14	recni 11233	. . . . . . . . . . . . 13 ⊢ γ ∈ ℂ
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → γ ∈ ℂ)
79	38, 38, 78	adddid 11243	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
80	12	recnd 11247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝐴)↑2) ∈ ℂ)
81	80	2halvesd 12463	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴)↑2))
82	38	sqvald 14113	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝐴)↑2) = ((log‘𝐴) · (log‘𝐴)))
83	81, 82	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) = ((log‘𝐴) · (log‘𝐴)))
84	83	oveq1d 7427	. . . . . . . . . . 11 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((log‘𝐴) · (log‘𝐴)) + ((log‘𝐴) · γ)))
85	73	recnd 11247	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) · γ) ∈ ℂ)
86	50, 50, 85	add32d 11446	. . . . . . . . . . 11 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (((log‘𝐴)↑2) / 2)) + ((log‘𝐴) · γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
87	79, 84, 86	3eqtr2d 2777	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝐴) · ((log‘𝐴) + γ)) = (((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)))
88	87	oveq1d 7427	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) = ((((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)) + (((log‘𝐴)↑2) / 2)) − (((log‘𝐴)↑2) / 2)))
89		mulcom 11199	. . . . . . . . . . 11 ⊢ ((γ ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
90	77, 38, 89	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (γ · (log‘𝐴)) = ((log‘𝐴) · γ))
91	90	oveq2d 7428	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = ((((log‘𝐴)↑2) / 2) + ((log‘𝐴) · γ)))
92	76, 88, 91	3eqtr4rd 2782	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) = (((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)))
93	92	oveq1d 7427	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿))
94	90, 85	eqeltrd 2832	. . . . . . . 8 ⊢ (𝜑 → (γ · (log‘𝐴)) ∈ ℂ)
95	50, 94, 51	addsubassd 11596	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝐴)↑2) / 2) + (γ · (log‘𝐴))) − 𝐿) = ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))
96	42, 50, 51	subsub4d 11607	. . . . . . 7 ⊢ (𝜑 → ((((log‘𝐴) · ((log‘𝐴) + γ)) − (((log‘𝐴)↑2) / 2)) − 𝐿) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
97	93, 95, 96	3eqtr3d 2779	. . . . . 6 ⊢ (𝜑 → ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)) = (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
98	71, 97	oveq12d 7430	. . . . 5 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)) − (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
99	37, 49, 42, 52	sub4d 11625	. . . . 5 ⊢ (𝜑 → ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚)) − (((log‘𝐴) · ((log‘𝐴) + γ)) − ((((log‘𝐴)↑2) / 2) + 𝐿))) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) − (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
100	98, 99	eqtrd 2771	. . . 4 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿))) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) − (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
101	100	fveq2d 6896	. . 3 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) = (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) − (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))))
102	43, 53	abs2dif2d 15410	. . 3 ⊢ (𝜑 → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) − (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))) ≤ ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))))
103	101, 102	eqbrtrd 5171	. 2 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ≤ ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))))
104		harmonicbnd4 26748	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))
105	2, 104	syl 17	. . . . . 6 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))
106	8	nnrecred 12268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
107	1, 106	fsumrecl 15685	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
108	107, 40	resubcld 11647	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℝ)
109	108	recnd 11247	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
110	109	abscld 15388	. . . . . . 7 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ)
111	2	rprecred 13032	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
112		0red 11222	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
113		1red 11220	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
114		0lt1 11741	. . . . . . . . 9 ⊢ 0 < 1
115	114	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 1)
116		loge 26328	. . . . . . . . 9 ⊢ (log‘e) = 1
117		mulog2sumlem1.3	. . . . . . . . . 10 ⊢ (𝜑 → e ≤ 𝐴)
118		epr 16156	. . . . . . . . . . 11 ⊢ e ∈ ℝ⁺
119		logleb 26344	. . . . . . . . . . 11 ⊢ ((e ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ⁺) → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
120	118, 2, 119	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (e ≤ 𝐴 ↔ (log‘e) ≤ (log‘𝐴)))
121	117, 120	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (log‘e) ≤ (log‘𝐴))
122	116, 121	eqbrtrrid 5185	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝐴))
123	112, 113, 11, 115, 122	ltletrd 11379	. . . . . . 7 ⊢ (𝜑 → 0 < (log‘𝐴))
124		lemul2 12072	. . . . . . 7 ⊢ (((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ ∧ ((log‘𝐴) ∈ ℝ ∧ 0 < (log‘𝐴))) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴) ↔ ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴))))
125	110, 111, 11, 123, 124	syl112anc 1373	. . . . . 6 ⊢ (𝜑 → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴) ↔ ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴))))
126	105, 125	mpbid 231	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) · (1 / 𝐴)))
127	45	rpcnd 13023	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℂ)
128	45	rpne0d 13026	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ≠ 0)
129	63, 127, 128	divrecd 11998	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → ((log‘𝐴) / 𝑚) = ((log‘𝐴) · (1 / 𝑚)))
130	129	sumeq2dv 15654	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
131	106	recnd 11247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℂ)
132	1, 38, 131	fsummulc2 15735	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) · (1 / 𝑚)))
133	130, 132	eqtr4d 2774	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) = ((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
134	133	oveq1d 7427	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = (((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) − ((log‘𝐴) · ((log‘𝐴) + γ))))
135	1, 131	fsumcl 15684	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
136	38, 135, 41	subdid 11675	. . . . . . . 8 ⊢ (𝜑 → ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) = (((log‘𝐴) · Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) − ((log‘𝐴) · ((log‘𝐴) + γ))))
137	134, 136	eqtr4d 2774	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ))) = ((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
138	137	fveq2d 6896	. . . . . 6 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) = (abs‘((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))))
139	135, 41	subcld 11576	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)) ∈ ℂ)
140	38, 139	absmuld 15406	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘𝐴) · (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) = ((abs‘(log‘𝐴)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))))
141	112, 11, 123	ltled 11367	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘𝐴))
142	11, 141	absidd 15374	. . . . . . 7 ⊢ (𝜑 → (abs‘(log‘𝐴)) = (log‘𝐴))
143	142	oveq1d 7427	. . . . . 6 ⊢ (𝜑 → ((abs‘(log‘𝐴)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))) = ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))))
144	138, 140, 143	3eqtrd 2775	. . . . 5 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) = ((log‘𝐴) · (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))))
145	2	rpcnd 13023	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
146	2	rpne0d 13026	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0)
147	38, 145, 146	divrecd 11998	. . . . 5 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) = ((log‘𝐴) · (1 / 𝐴)))
148	126, 144, 147	3brtr4d 5181	. . . 4 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) ≤ ((log‘𝐴) / 𝐴))
149		fveq2 6892	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
150		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → 𝑖 = 𝑚)
151	149, 150	oveq12d 7430	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → ((log‘𝑖) / 𝑖) = ((log‘𝑚) / 𝑚))
152	151	cbvsumv 15647	. . . . . . . . . . . 12 ⊢ Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚)
153		fveq2 6892	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (⌊‘𝑦) = (⌊‘𝐴))
154	153	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (1...(⌊‘𝑦)) = (1...(⌊‘𝐴)))
155	154	sumeq1d 15652	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → Σ𝑚 ∈ (1...(⌊‘𝑦))((log‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
156	152, 155	eqtrid 2783	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚))
157		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (log‘𝑦) = (log‘𝐴))
158	157	oveq1d 7427	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ((log‘𝑦)↑2) = ((log‘𝐴)↑2))
159	158	oveq1d 7427	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (((log‘𝑦)↑2) / 2) = (((log‘𝐴)↑2) / 2))
160	156, 159	oveq12d 7430	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (Σ𝑖 ∈ (1...(⌊‘𝑦))((log‘𝑖) / 𝑖) − (((log‘𝑦)↑2) / 2)) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)))
161		ovex 7445	. . . . . . . . . 10 ⊢ (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) ∈ V
162	160, 19, 161	fvmpt 6999	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐹‘𝐴) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)))
163	2, 162	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)))
164	163	oveq1d 7427	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿))
165	49, 50, 51	subsub4d 11607	. . . . . . 7 ⊢ (𝜑 → ((Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − (((log‘𝐴)↑2) / 2)) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
166	164, 165	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴) − 𝐿) = (Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))
167	166	fveq2d 6896	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) − 𝐿)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))))
168	20	simp3i 1140	. . . . . 6 ⊢ ((𝐹 ⇝_𝑟 𝐿 ∧ 𝐴 ∈ ℝ⁺ ∧ e ≤ 𝐴) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
169	24, 2, 117, 168	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ ((log‘𝐴) / 𝐴))
170	167, 169	eqbrtrrd 5173	. . . 4 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿))) ≤ ((log‘𝐴) / 𝐴))
171	44, 54, 57, 57, 148, 170	le2addd 11838	. . 3 ⊢ (𝜑 → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))) ≤ (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
172	57	recnd 11247	. . . 4 ⊢ (𝜑 → ((log‘𝐴) / 𝐴) ∈ ℂ)
173	172	2timesd 12460	. . 3 ⊢ (𝜑 → (2 · ((log‘𝐴) / 𝐴)) = (((log‘𝐴) / 𝐴) + ((log‘𝐴) / 𝐴)))
174	171, 173	breqtrrd 5177	. 2 ⊢ (𝜑 → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝐴) / 𝑚) − ((log‘𝐴) · ((log‘𝐴) + γ)))) + (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘𝑚) / 𝑚) − ((((log‘𝐴)↑2) / 2) + 𝐿)))) ≤ (2 · ((log‘𝐴) / 𝐴)))
175	33, 55, 59, 103, 174	letrd 11376	1 ⊢ (𝜑 → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))((log‘(𝐴 / 𝑚)) / 𝑚) − ((((log‘𝐴)↑2) / 2) + ((γ · (log‘𝐴)) − 𝐿)))) ≤ (2 · ((log‘𝐴) / 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 supcsup 9438 ℂcc 11111 ℝcr 11112 0cc0 11113 1c1 11114 + caddc 11116 · cmul 11118 +∞cpnf 11250 ℝ^*cxr 11252 < clt 11253 ≤ cle 11254 − cmin 11449 / cdiv 11876 ℕcn 12217 2c2 12272 ℝ⁺crp 12979 ...cfz 13489 ⌊cfl 13760 ↑cexp 14032 abscabs 15186 ⇝_𝑟 crli 15434 Σcsu 15637 eceu 16011 logclog 26296 γcem 26729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-oadd 8473 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-xnn0 12550 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-e 16017 df-sin 16018 df-cos 16019 df-tan 16020 df-pi 16021 df-dvds 16203 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-cmp 23112 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617 df-ulm 26122 df-log 26298 df-cxp 26299 df-atan 26605 df-em 26730

This theorem is referenced by: mulog2sumlem2 27271

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