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Theorem log2sumbnd 27044

Description: Bound on the difference between Σ𝑛 ≤ 𝐴, log↑2(𝑛) and the equivalent integral. (Contributed by Mario Carneiro, 20-May-2016.)

**Assertion**
Ref	Expression
log2sumbnd	⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ≤ (((log‘𝐴)↑2) + 2))

Distinct variable group: 𝐴,𝑛

Proof of Theorem log2sumbnd Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13937	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
2		elfznn 13529	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
3	2	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
4	3	nnrpd 13013	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
5	4	relogcld 26130	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
6	5	resqcld 14089	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛)↑2) ∈ ℝ)
7	1, 6	fsumrecl 15679	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) ∈ ℝ)
8		rpre 12981	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
9	8	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
10		relogcl 26083	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
11	10	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
12	11	resqcld 14089	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘𝐴)↑2) ∈ ℝ)
13		2re 12285	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
14		remulcl 11194	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (2 · (log‘𝐴)) ∈ ℝ)
15	13, 11, 14	sylancr 587	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (2 · (log‘𝐴)) ∈ ℝ)
16		resubcl 11523	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ (2 · (log‘𝐴)) ∈ ℝ) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
17	13, 15, 16	sylancr 587	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
18	12, 17	readdcld 11242	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))) ∈ ℝ)
19	9, 18	remulcld 11243	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))) ∈ ℝ)
20	7, 19	resubcld 11641	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℝ)
21	20	recnd 11241	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ)
22	21	abscld 15382	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ)
23		resubcl 11523	. . . 4 ⊢ (((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ ∧ 2 ∈ ℝ) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ∈ ℝ)
24	22, 13, 23	sylancl 586	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ∈ ℝ)
25		2cn 12286	. . . . . 6 ⊢ 2 ∈ ℂ
26	25	negcli 11527	. . . . 5 ⊢ -2 ∈ ℂ
27		subcl 11458	. . . . 5 ⊢ (((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ ∧ -2 ∈ ℂ) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2) ∈ ℂ)
28	21, 26, 27	sylancl 586	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2) ∈ ℂ)
29	28	abscld 15382	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ∈ ℝ)
30	25	absnegi 15346	. . . . . 6 ⊢ (abs‘-2) = (abs‘2)
31		0le2 12313	. . . . . . 7 ⊢ 0 ≤ 2
32		absid 15242	. . . . . . 7 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
33	13, 31, 32	mp2an 690	. . . . . 6 ⊢ (abs‘2) = 2
34	30, 33	eqtri 2760	. . . . 5 ⊢ (abs‘-2) = 2
35	34	oveq2i 7419	. . . 4 ⊢ ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − (abs‘-2)) = ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2)
36		abs2dif 15278	. . . . 5 ⊢ (((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ ∧ -2 ∈ ℂ) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − (abs‘-2)) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)))
37	21, 26, 36	sylancl 586	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − (abs‘-2)) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)))
38	35, 37	eqbrtrrid 5184	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)))
39		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
40	39	oveq2d 7424	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
41	40	sumeq1d 15646	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2))
42		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
43		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
44	43	oveq1d 7423	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((log‘𝑥)↑2) = ((log‘𝐴)↑2))
45	43	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (2 · (log‘𝑥)) = (2 · (log‘𝐴)))
46	45	oveq2d 7424	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (2 − (2 · (log‘𝑥))) = (2 − (2 · (log‘𝐴))))
47	44, 46	oveq12d 7426	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))
48	42, 47	oveq12d 7426	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))
49	41, 48	oveq12d 7426	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))))
50		eqid 2732	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))
51		ovex 7441	. . . . . . . 8 ⊢ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) ∈ V
52	49, 50, 51	fvmpt3i 7003	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))))
53	52	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))))
54		1rp 12977	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
55		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
56		1z 12591	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
57		flid 13772	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → (⌊‘1) = 1)
58	56, 57	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (⌊‘1) = 1
59	55, 58	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (⌊‘𝑥) = 1)
60	59	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
61	60	sumeq1d 15646	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...1)((log‘𝑛)↑2))
62		0cn 11205	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
63		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
64		log1 26093	. . . . . . . . . . . . . . 15 ⊢ (log‘1) = 0
65	63, 64	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
66	65	sq0id 14157	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → ((log‘𝑛)↑2) = 0)
67	66	fsum1 15692	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0)
68	56, 62, 67	mp2an 690	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0
69	61, 68	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = 0)
70		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → 𝑥 = 1)
71		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (log‘𝑥) = (log‘1))
72	71, 64	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → (log‘𝑥) = 0)
73	72	sq0id 14157	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → ((log‘𝑥)↑2) = 0)
74	72	oveq2d 7424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → (2 · (log‘𝑥)) = (2 · 0))
75		2t0e0 12380	. . . . . . . . . . . . . . . . 17 ⊢ (2 · 0) = 0
76	74, 75	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (2 · (log‘𝑥)) = 0)
77	76	oveq2d 7424	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = (2 − 0))
78	25	subid1i 11531	. . . . . . . . . . . . . . 15 ⊢ (2 − 0) = 2
79	77, 78	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = 2)
80	73, 79	oveq12d 7426	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (0 + 2))
81	25	addlidi 11401	. . . . . . . . . . . . 13 ⊢ (0 + 2) = 2
82	80, 81	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = 2)
83	70, 82	oveq12d 7426	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (1 · 2))
84	25	mullidi 11218	. . . . . . . . . . 11 ⊢ (1 · 2) = 2
85	83, 84	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = 2)
86	69, 85	oveq12d 7426	. . . . . . . . 9 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (0 − 2))
87		df-neg 11446	. . . . . . . . 9 ⊢ -2 = (0 − 2)
88	86, 87	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = -2)
89	88, 50, 51	fvmpt3i 7003	. . . . . . 7 ⊢ (1 ∈ ℝ⁺ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1) = -2)
90	54, 89	mp1i 13	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1) = -2)
91	53, 90	oveq12d 7426	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) − ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1)) = ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2))
92	91	fveq2d 6895	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) − ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1))) = (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)))
93		ioorp 13401	. . . . . 6 ⊢ (0(,)+∞) = ℝ⁺
94	93	eqcomi 2741	. . . . 5 ⊢ ℝ⁺ = (0(,)+∞)
95		nnuz 12864	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
96	56	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
97		1red 11214	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
98		pnfxr 11267	. . . . . 6 ⊢ +∞ ∈ ℝ^*
99	98	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ^*)
100		1re 11213	. . . . . . 7 ⊢ 1 ∈ ℝ
101		1nn0 12487	. . . . . . 7 ⊢ 1 ∈ ℕ₀
102	100, 101	nn0addge1i 12519	. . . . . 6 ⊢ 1 ≤ (1 + 1)
103	102	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
104		0red 11216	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
105		rpre 12981	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
106	105	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
107		simpr 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
108	107	relogcld 26130	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
109	108	resqcld 14089	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥)↑2) ∈ ℝ)
110		remulcl 11194	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → (2 · (log‘𝑥)) ∈ ℝ)
111	13, 108, 110	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (log‘𝑥)) ∈ ℝ)
112		resubcl 11523	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (2 · (log‘𝑥)) ∈ ℝ) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
113	13, 111, 112	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
114	109, 113	readdcld 11242	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℝ)
115	106, 114	remulcld 11243	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) ∈ ℝ)
116		nnrp 12984	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
117	116, 109	sylan2 593	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → ((log‘𝑥)↑2) ∈ ℝ)
118		reelprrecn 11201	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
119	118	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
120	106	recnd 11241	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
121		1red 11214	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
122		recn 11199	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
123	122	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
124		1red 11214	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
125	119	dvmptid 25473	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
126		rpssre 12980	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
127	126	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ℝ⁺ ⊆ ℝ)
128		eqid 2732	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
129	128	tgioo2 24318	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
130		iooretop 24281	. . . . . . . . . 10 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
131	93, 130	eqeltrri 2830	. . . . . . . . 9 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
132	131	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ℝ⁺ ∈ (topGen‘ran (,)))
133	119, 123, 124, 125, 127, 129, 128, 132	dvmptres 25479	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ 1))
134	114	recnd 11241	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℂ)
135		resubcl 11523	. . . . . . . . 9 ⊢ (((2 · (log‘𝑥)) ∈ ℝ ∧ 2 ∈ ℝ) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
136	111, 13, 135	sylancl 586	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
137	136, 107	rerpdivcld 13046	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) − 2) / 𝑥) ∈ ℝ)
138	109	recnd 11241	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥)↑2) ∈ ℂ)
139	111	recnd 11241	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (log‘𝑥)) ∈ ℂ)
140	107	rpreccld 13025	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
141	140	rpcnd 13017	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
142	139, 141	mulcld 11233	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (log‘𝑥)) · (1 / 𝑥)) ∈ ℂ)
143		cnelprrecn 11202	. . . . . . . . . . 11 ⊢ ℂ ∈ {ℝ, ℂ}
144	143	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ℂ ∈ {ℝ, ℂ})
145	108	recnd 11241	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
146		sqcl 14082	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
147	146	adantl 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
148		simpr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
149		mulcl 11193	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
150	25, 148, 149	sylancr 587	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
151		relogf1o 26074	. . . . . . . . . . . . . . 15 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
152		f1of 6833	. . . . . . . . . . . . . . 15 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
153	151, 152	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
154	153	feqmptd 6960	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
155		fvres 6910	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
156	155	mpteq2ia 5251	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
157	154, 156	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
158	157	oveq2d 7424	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
159		dvrelog 26144	. . . . . . . . . . 11 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
160	158, 159	eqtr3di 2787	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
161		2nn 12284	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
162		dvexp 25469	. . . . . . . . . . . 12 ⊢ (2 ∈ ℕ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))))
163	161, 162	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))))
164		2m1e1 12337	. . . . . . . . . . . . . . 15 ⊢ (2 − 1) = 1
165	164	oveq2i 7419	. . . . . . . . . . . . . 14 ⊢ (𝑦↑(2 − 1)) = (𝑦↑1)
166		exp1 14032	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℂ → (𝑦↑1) = 𝑦)
167	165, 166	eqtrid 2784	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (𝑦↑(2 − 1)) = 𝑦)
168	167	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (2 · (𝑦↑(2 − 1))) = (2 · 𝑦))
169	168	mpteq2ia 5251	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦))
170	163, 169	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦)))
171		oveq1 7415	. . . . . . . . . 10 ⊢ (𝑦 = (log‘𝑥) → (𝑦↑2) = ((log‘𝑥)↑2))
172		oveq2 7416	. . . . . . . . . 10 ⊢ (𝑦 = (log‘𝑥) → (2 · 𝑦) = (2 · (log‘𝑥)))
173	119, 144, 145, 140, 147, 150, 160, 170, 171, 172	dvmptco 25488	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥)↑2))) = (𝑥 ∈ ℝ⁺ ↦ ((2 · (log‘𝑥)) · (1 / 𝑥))))
174	113	recnd 11241	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 − (2 · (log‘𝑥))) ∈ ℂ)
175		ovexd 7443	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (0 − (2 · (1 / 𝑥))) ∈ V)
176		2cnd 12289	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
177		0red 11216	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ∈ ℝ)
178		2cnd 12289	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 2 ∈ ℂ)
179		0red 11216	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
180		2cnd 12289	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 2 ∈ ℂ)
181	119, 180	dvmptc 25474	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 2)) = (𝑥 ∈ ℝ ↦ 0))
182	119, 178, 179, 181, 127, 129, 128, 132	dvmptres 25479	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 2)) = (𝑥 ∈ ℝ⁺ ↦ 0))
183		mulcl 11193	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (1 / 𝑥) ∈ ℂ) → (2 · (1 / 𝑥)) ∈ ℂ)
184	25, 141, 183	sylancr 587	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (1 / 𝑥)) ∈ ℂ)
185	119, 145, 140, 160, 180	dvmptcmul 25480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (2 · (1 / 𝑥))))
186	119, 176, 177, 182, 139, 184, 185	dvmptsub 25483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (2 − (2 · (log‘𝑥))))) = (𝑥 ∈ ℝ⁺ ↦ (0 − (2 · (1 / 𝑥)))))
187	119, 138, 142, 173, 174, 175, 186	dvmptadd 25476	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (𝑥 ∈ ℝ⁺ ↦ (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))))
188	139, 176, 141	subdird 11670	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
189	136	recnd 11241	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (log‘𝑥)) − 2) ∈ ℂ)
190		rpne0 12989	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
191	190	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
192	189, 120, 191	divrecd 11992	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) − 2) / 𝑥) = (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)))
193		df-neg 11446	. . . . . . . . . . . 12 ⊢ -(2 · (1 / 𝑥)) = (0 − (2 · (1 / 𝑥)))
194	193	oveq2i 7419	. . . . . . . . . . 11 ⊢ (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))
195	142, 184	negsubd 11576	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
196	194, 195	eqtr3id 2786	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
197	188, 192, 196	3eqtr4rd 2783	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) − 2) / 𝑥))
198	197	mpteq2dva 5248	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝑥 ∈ ℝ⁺ ↦ (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))) = (𝑥 ∈ ℝ⁺ ↦ (((2 · (log‘𝑥)) − 2) / 𝑥)))
199	187, 198	eqtrd 2772	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (𝑥 ∈ ℝ⁺ ↦ (((2 · (log‘𝑥)) − 2) / 𝑥)))
200	119, 120, 121, 133, 134, 137, 199	dvmptmul 25477	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ⁺ ↦ ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥))))
201	134	mullidd 11231	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
202	138, 139, 176	subsub2d 11599	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
203	201, 202	eqtr4d 2775	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)))
204	189, 120, 191	divcan1d 11990	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥) = ((2 · (log‘𝑥)) − 2))
205	203, 204	oveq12d 7426	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥)) = ((((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) + ((2 · (log‘𝑥)) − 2)))
206	138, 189	npcand 11574	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) + ((2 · (log‘𝑥)) − 2)) = ((log‘𝑥)↑2))
207	205, 206	eqtrd 2772	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥)) = ((log‘𝑥)↑2))
208	207	mpteq2dva 5248	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝑥 ∈ ℝ⁺ ↦ ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥)↑2)))
209	200, 208	eqtrd 2772	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥)↑2)))
210		fveq2 6891	. . . . . 6 ⊢ (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
211	210	oveq1d 7423	. . . . 5 ⊢ (𝑥 = 𝑛 → ((log‘𝑥)↑2) = ((log‘𝑛)↑2))
212		simp32 1210	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ≤ 𝑛)
213		simp2l 1199	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ∈ ℝ⁺)
214		simp2r 1200	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑛 ∈ ℝ⁺)
215	213, 214	logled 26134	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (𝑥 ≤ 𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
216	212, 215	mpbid 231	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘𝑥) ≤ (log‘𝑛))
217	213	relogcld 26130	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘𝑥) ∈ ℝ)
218	214	relogcld 26130	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘𝑛) ∈ ℝ)
219		simp31 1209	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 1 ≤ 𝑥)
220		logleb 26110	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
221	54, 213, 220	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
222	219, 221	mpbid 231	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘1) ≤ (log‘𝑥))
223	64, 222	eqbrtrrid 5184	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 0 ≤ (log‘𝑥))
224	214	rpred 13015	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑛 ∈ ℝ)
225		1red 11214	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 1 ∈ ℝ)
226	213	rpred 13015	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ∈ ℝ)
227	225, 226, 224, 219, 212	letrd 11370	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 1 ≤ 𝑛)
228	224, 227	logge0d 26137	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 0 ≤ (log‘𝑛))
229	217, 218, 223, 228	le2sqd 14219	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → ((log‘𝑥) ≤ (log‘𝑛) ↔ ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2)))
230	216, 229	mpbid 231	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2))
231		relogcl 26083	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
232	231	ad2antrl 726	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
233	232	sqge0d 14101	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥)↑2))
234	54	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ⁺)
235		simpl 483	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
236		1le1 11841	. . . . . 6 ⊢ 1 ≤ 1
237	236	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
238		simpr 485	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
239	9	rexrd 11263	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ^*)
240		pnfge 13109	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
241	239, 240	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ≤ +∞)
242	94, 95, 96, 97, 99, 103, 104, 115, 109, 117, 209, 211, 230, 50, 233, 234, 235, 237, 238, 241, 44	dvfsum2 25550	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) − ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1))) ≤ ((log‘𝐴)↑2))
243	92, 242	eqbrtrrd 5172	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ≤ ((log‘𝐴)↑2))
244	24, 29, 12, 38, 243	letrd 11370	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ≤ ((log‘𝐴)↑2))
245	13	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 2 ∈ ℝ)
246	22, 245, 12	lesubaddd 11810	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ≤ ((log‘𝐴)↑2) ↔ (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ≤ (((log‘𝐴)↑2) + 2)))
247	244, 246	mpbid 231	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ≤ (((log‘𝐴)↑2) + 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ⊆ wss 3948 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 ran crn 5677 ↾ cres 5678 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 +∞cpnf 11244 ℝ^*cxr 11246 ≤ cle 11248 − cmin 11443 -cneg 11444 / cdiv 11870 ℕcn 12211 2c2 12266 ℤcz 12557 ℝ⁺crp 12973 (,)cioo 13323 ...cfz 13483 ⌊cfl 13754 ↑cexp 14026 abscabs 15180 Σcsu 15631 TopOpenctopn 17366 topGenctg 17382 ℂ_fldccnfld 20943 D cdv 25379 logclog 26062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-cmp 22890 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cncf 24393 df-limc 25382 df-dv 25383 df-log 26064

This theorem is referenced by: selberglem2 27046

W3C validator