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Theorem log2sumbnd 27475

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 13872	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
2		elfznn 13445	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
3	2	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
4	3	nnrpd 12924	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
5	4	relogcld 26552	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
6	5	resqcld 14024	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛)↑2) ∈ ℝ)
7	1, 6	fsumrecl 15633	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) ∈ ℝ)
8		rpre 12891	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
10		relogcl 26504	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
11	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
12	11	resqcld 14024	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘𝐴)↑2) ∈ ℝ)
13		2re 12191	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
14		remulcl 11083	. . . . . . . . . . 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 11417	. . . . . . . . . 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 11133	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))) ∈ ℝ)
19	9, 18	remulcld 11134	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))) ∈ ℝ)
20	7, 19	resubcld 11537	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℝ)
21	20	recnd 11132	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ)
22	21	abscld 15338	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ)
23		resubcl 11417	. . . 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 12192	. . . . . 6 ⊢ 2 ∈ ℂ
26	25	negcli 11421	. . . . 5 ⊢ -2 ∈ ℂ
27		subcl 11351	. . . . 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 15338	. . 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 15300	. . . . . 6 ⊢ (abs‘-2) = (abs‘2)
31		0le2 12219	. . . . . . 7 ⊢ 0 ≤ 2
32		absid 15195	. . . . . . 7 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
33	13, 31, 32	mp2an 692	. . . . . 6 ⊢ (abs‘2) = 2
34	30, 33	eqtri 2753	. . . . 5 ⊢ (abs‘-2) = 2
35	34	oveq2i 7352	. . . 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 15232	. . . . 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 5125	. . 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 6817	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
40	39	oveq2d 7357	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
41	40	sumeq1d 15599	. . . . . . . . 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 6817	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
44	43	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((log‘𝑥)↑2) = ((log‘𝐴)↑2))
45	43	oveq2d 7357	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (2 · (log‘𝑥)) = (2 · (log‘𝐴)))
46	45	oveq2d 7357	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (2 − (2 · (log‘𝑥))) = (2 − (2 · (log‘𝐴))))
47	44, 46	oveq12d 7359	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))
48	42, 47	oveq12d 7359	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))
49	41, 48	oveq12d 7359	. . . . . . . 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 2730	. . . . . . . 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 7374	. . . . . . . 8 ⊢ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) ∈ V
52	49, 50, 51	fvmpt3i 6929	. . . . . . 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 480	. . . . . 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 12886	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
55		fveq2 6817	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
56		1z 12494	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
57		flid 13704	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → (⌊‘1) = 1)
58	56, 57	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (⌊‘1) = 1
59	55, 58	eqtrdi 2781	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (⌊‘𝑥) = 1)
60	59	oveq2d 7357	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
61	60	sumeq1d 15599	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...1)((log‘𝑛)↑2))
62		0cn 11096	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
63		fveq2 6817	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
64		log1 26514	. . . . . . . . . . . . . . 15 ⊢ (log‘1) = 0
65	63, 64	eqtrdi 2781	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
66	65	sq0id 14093	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → ((log‘𝑛)↑2) = 0)
67	66	fsum1 15646	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0)
68	56, 62, 67	mp2an 692	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0
69	61, 68	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = 0)
70		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → 𝑥 = 1)
71		fveq2 6817	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (log‘𝑥) = (log‘1))
72	71, 64	eqtrdi 2781	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → (log‘𝑥) = 0)
73	72	sq0id 14093	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → ((log‘𝑥)↑2) = 0)
74	72	oveq2d 7357	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → (2 · (log‘𝑥)) = (2 · 0))
75		2t0e0 12281	. . . . . . . . . . . . . . . . 17 ⊢ (2 · 0) = 0
76	74, 75	eqtrdi 2781	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (2 · (log‘𝑥)) = 0)
77	76	oveq2d 7357	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = (2 − 0))
78	25	subid1i 11425	. . . . . . . . . . . . . . 15 ⊢ (2 − 0) = 2
79	77, 78	eqtrdi 2781	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = 2)
80	73, 79	oveq12d 7359	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (0 + 2))
81	25	addlidi 11293	. . . . . . . . . . . . 13 ⊢ (0 + 2) = 2
82	80, 81	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = 2)
83	70, 82	oveq12d 7359	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (1 · 2))
84	25	mullidi 11109	. . . . . . . . . . 11 ⊢ (1 · 2) = 2
85	83, 84	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = 2)
86	69, 85	oveq12d 7359	. . . . . . . . 9 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (0 − 2))
87		df-neg 11339	. . . . . . . . 9 ⊢ -2 = (0 − 2)
88	86, 87	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = -2)
89	88, 50, 51	fvmpt3i 6929	. . . . . . 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 7359	. . . . 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 6821	. . . 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 13317	. . . . . 6 ⊢ (0(,)+∞) = ℝ⁺
94	93	eqcomi 2739	. . . . 5 ⊢ ℝ⁺ = (0(,)+∞)
95		nnuz 12767	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
96	56	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
97		1red 11105	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
98		pnfxr 11158	. . . . . 6 ⊢ +∞ ∈ ℝ^*
99	98	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ^*)
100		1re 11104	. . . . . . 7 ⊢ 1 ∈ ℝ
101		1nn0 12389	. . . . . . 7 ⊢ 1 ∈ ℕ₀
102	100, 101	nn0addge1i 12421	. . . . . 6 ⊢ 1 ≤ (1 + 1)
103	102	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
104		0red 11107	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
105		rpre 12891	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
106	105	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
107		simpr 484	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
108	107	relogcld 26552	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
109	108	resqcld 14024	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥)↑2) ∈ ℝ)
110		remulcl 11083	. . . . . . . . 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 11417	. . . . . . . 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 11133	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℝ)
115	106, 114	remulcld 11134	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) ∈ ℝ)
116		nnrp 12894	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
117	116, 109	sylan2 593	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → ((log‘𝑥)↑2) ∈ ℝ)
118		reelprrecn 11090	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
119	118	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
120	106	recnd 11132	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
121		1red 11105	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
122		recn 11088	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
123	122	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
124		1red 11105	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
125	119	dvmptid 25881	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
126		rpssre 12890	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
127	126	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ℝ⁺ ⊆ ℝ)
128		tgioo4 24713	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
129		eqid 2730	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
130		iooretop 24673	. . . . . . . . . 10 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
131	93, 130	eqeltrri 2826	. . . . . . . . 9 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
132	131	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ℝ⁺ ∈ (topGen‘ran (,)))
133	119, 123, 124, 125, 127, 128, 129, 132	dvmptres 25887	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ 1))
134	114	recnd 11132	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℂ)
135		resubcl 11417	. . . . . . . . 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 12957	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) − 2) / 𝑥) ∈ ℝ)
138	109	recnd 11132	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥)↑2) ∈ ℂ)
139	111	recnd 11132	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (log‘𝑥)) ∈ ℂ)
140	107	rpreccld 12936	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
141	140	rpcnd 12928	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
142	139, 141	mulcld 11124	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (log‘𝑥)) · (1 / 𝑥)) ∈ ℂ)
143		cnelprrecn 11091	. . . . . . . . . . 11 ⊢ ℂ ∈ {ℝ, ℂ}
144	143	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ℂ ∈ {ℝ, ℂ})
145	108	recnd 11132	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
146		sqcl 14017	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
147	146	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
148		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
149		mulcl 11082	. . . . . . . . . . 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 26495	. . . . . . . . . . . . . . 15 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
152		f1of 6759	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
155		fvres 6836	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
156	155	mpteq2ia 5184	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
157	154, 156	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
158	157	oveq2d 7357	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
159		dvrelog 26566	. . . . . . . . . . 11 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
160	158, 159	eqtr3di 2780	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
161		2nn 12190	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
162		dvexp 25877	. . . . . . . . . . . 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 12238	. . . . . . . . . . . . . . 15 ⊢ (2 − 1) = 1
165	164	oveq2i 7352	. . . . . . . . . . . . . 14 ⊢ (𝑦↑(2 − 1)) = (𝑦↑1)
166		exp1 13966	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℂ → (𝑦↑1) = 𝑦)
167	165, 166	eqtrid 2777	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (𝑦↑(2 − 1)) = 𝑦)
168	167	oveq2d 7357	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (2 · (𝑦↑(2 − 1))) = (2 · 𝑦))
169	168	mpteq2ia 5184	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦))
170	163, 169	eqtrdi 2781	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦)))
171		oveq1 7348	. . . . . . . . . 10 ⊢ (𝑦 = (log‘𝑥) → (𝑦↑2) = ((log‘𝑥)↑2))
172		oveq2 7349	. . . . . . . . . 10 ⊢ (𝑦 = (log‘𝑥) → (2 · 𝑦) = (2 · (log‘𝑥)))
173	119, 144, 145, 140, 147, 150, 160, 170, 171, 172	dvmptco 25896	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥)↑2))) = (𝑥 ∈ ℝ⁺ ↦ ((2 · (log‘𝑥)) · (1 / 𝑥))))
174	113	recnd 11132	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 − (2 · (log‘𝑥))) ∈ ℂ)
175		ovexd 7376	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (0 − (2 · (1 / 𝑥))) ∈ V)
176		2cnd 12195	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
177		0red 11107	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ∈ ℝ)
178		2cnd 12195	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 2 ∈ ℂ)
179		0red 11107	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
180		2cnd 12195	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 2 ∈ ℂ)
181	119, 180	dvmptc 25882	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 2)) = (𝑥 ∈ ℝ ↦ 0))
182	119, 178, 179, 181, 127, 128, 129, 132	dvmptres 25887	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 2)) = (𝑥 ∈ ℝ⁺ ↦ 0))
183		mulcl 11082	. . . . . . . . . . 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 25888	. . . . . . . . . 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 25891	. . . . . . . . 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 25884	. . . . . . . 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 11566	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
189	136	recnd 11132	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (log‘𝑥)) − 2) ∈ ℂ)
190		rpne0 12899	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
191	190	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
192	189, 120, 191	divrecd 11892	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) − 2) / 𝑥) = (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)))
193		df-neg 11339	. . . . . . . . . . . 12 ⊢ -(2 · (1 / 𝑥)) = (0 − (2 · (1 / 𝑥)))
194	193	oveq2i 7352	. . . . . . . . . . 11 ⊢ (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))
195	142, 184	negsubd 11470	. . . . . . . . . . 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 2779	. . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) − 2) / 𝑥))
198	197	mpteq2dva 5182	. . . . . . . 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 2765	. . . . . . 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 25885	. . . . . 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 11122	. . . . . . . . . 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 11493	. . . . . . . . . 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 2768	. . . . . . . . 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 11890	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥) = ((2 · (log‘𝑥)) − 2))
205	203, 204	oveq12d 7359	. . . . . . . 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 11468	. . . . . . . 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 2765	. . . . . . 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 5182	. . . . . 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 2765	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥)↑2)))
210		fveq2 6817	. . . . . 6 ⊢ (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
211	210	oveq1d 7356	. . . . 5 ⊢ (𝑥 = 𝑛 → ((log‘𝑥)↑2) = ((log‘𝑛)↑2))
212		simp32 1211	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ≤ 𝑛)
213		simp2l 1200	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ∈ ℝ⁺)
214		simp2r 1201	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑛 ∈ ℝ⁺)
215	213, 214	logled 26556	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (𝑥 ≤ 𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
216	212, 215	mpbid 232	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘𝑥) ≤ (log‘𝑛))
217	213	relogcld 26552	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘𝑥) ∈ ℝ)
218	214	relogcld 26552	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘𝑛) ∈ ℝ)
219		simp31 1210	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 1 ≤ 𝑥)
220		logleb 26532	. . . . . . . . . 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 232	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘1) ≤ (log‘𝑥))
223	64, 222	eqbrtrrid 5125	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 0 ≤ (log‘𝑥))
224	214	rpred 12926	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑛 ∈ ℝ)
225		1red 11105	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 1 ∈ ℝ)
226	213	rpred 12926	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ∈ ℝ)
227	225, 226, 224, 219, 212	letrd 11262	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 1 ≤ 𝑛)
228	224, 227	logge0d 26559	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 0 ≤ (log‘𝑛))
229	217, 218, 223, 228	le2sqd 14156	. . . . . 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 232	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2))
231		relogcl 26504	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
232	231	ad2antrl 728	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
233	232	sqge0d 14036	. . . . 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 482	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
236		1le1 11737	. . . . . 6 ⊢ 1 ≤ 1
237	236	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
238		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
239	9	rexrd 11154	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ^*)
240		pnfge 13021	. . . . . 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 25961	. . . 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 5113	. . 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 11262	. 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 11706	. 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 232	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 ≠ wne 2926 Vcvv 3434 ⊆ wss 3900 {cpr 4576 class class class wbr 5089 ↦ cmpt 5170 ran crn 5615 ↾ cres 5616 ⟶wf 6473 –1-1-onto→wf1o 6476 ‘cfv 6477 (class class class)co 7341 ℂcc 10996 ℝcr 10997 0cc0 10998 1c1 10999 + caddc 11001 · cmul 11003 +∞cpnf 11135 ℝ^*cxr 11137 ≤ cle 11139 − cmin 11336 -cneg 11337 / cdiv 11766 ℕcn 12117 2c2 12172 ℤcz 12460 ℝ⁺crp 12882 (,)cioo 13237 ...cfz 13399 ⌊cfl 13686 ↑cexp 13960 abscabs 15133 Σcsu 15585 TopOpenctopn 17317 topGenctg 17333 ℂ_fldccnfld 21284 D cdv 25784 logclog 26483

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 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-sum 15586 df-ef 15966 df-sin 15968 df-cos 15969 df-pi 15971 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-submnd 18684 df-mulg 18973 df-cntz 19222 df-cmn 19687 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-cmp 23295 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-limc 25787 df-dv 25788 df-log 26485

This theorem is referenced by: selberglem2 27477

W3C validator