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Theorem rplogsumlem1 26994

Description: Lemma for rplogsum 27037. (Contributed by Mario Carneiro, 2-May-2016.)

**Assertion**
Ref	Expression
rplogsumlem1	⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2)

Distinct variable group: 𝐴,𝑛

Proof of Theorem rplogsumlem1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13940	. . 3 ⊢ (𝐴 ∈ ℕ → (2...𝐴) ∈ Fin)
2		elfzuz 13499	. . . . . . . 8 ⊢ (𝑛 ∈ (2...𝐴) → 𝑛 ∈ (ℤ_≥‘2))
3		eluz2nn 12870	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘2) → 𝑛 ∈ ℕ)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝑛 ∈ (2...𝐴) → 𝑛 ∈ ℕ)
5	4	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 𝑛 ∈ ℕ)
6	5	nnrpd 13016	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 𝑛 ∈ ℝ⁺)
7	6	relogcld 26138	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (log‘𝑛) ∈ ℝ)
8	2	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 𝑛 ∈ (ℤ_≥‘2))
9		uz2m1nn 12909	. . . . . 6 ⊢ (𝑛 ∈ (ℤ_≥‘2) → (𝑛 − 1) ∈ ℕ)
10	8, 9	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 − 1) ∈ ℕ)
11	5, 10	nnmulcld 12267	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 · (𝑛 − 1)) ∈ ℕ)
12	7, 11	nndivred 12268	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((log‘𝑛) / (𝑛 · (𝑛 − 1))) ∈ ℝ)
13	1, 12	fsumrecl 15682	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ∈ ℝ)
14		2re 12288	. . . . 5 ⊢ 2 ∈ ℝ
15	10	nnrpd 13016	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 − 1) ∈ ℝ⁺)
16	15	rpsqrtcld 15360	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘(𝑛 − 1)) ∈ ℝ⁺)
17		rerpdivcl 13006	. . . . 5 ⊢ ((2 ∈ ℝ ∧ (√‘(𝑛 − 1)) ∈ ℝ⁺) → (2 / (√‘(𝑛 − 1))) ∈ ℝ)
18	14, 16, 17	sylancr 587	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (2 / (√‘(𝑛 − 1))) ∈ ℝ)
19	6	rpsqrtcld 15360	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘𝑛) ∈ ℝ⁺)
20		rerpdivcl 13006	. . . . 5 ⊢ ((2 ∈ ℝ ∧ (√‘𝑛) ∈ ℝ⁺) → (2 / (√‘𝑛)) ∈ ℝ)
21	14, 19, 20	sylancr 587	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (2 / (√‘𝑛)) ∈ ℝ)
22	18, 21	resubcld 11644	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) ∈ ℝ)
23	1, 22	fsumrecl 15682	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) ∈ ℝ)
24	14	a1i 11	. 2 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℝ)
25	16	rpred 13018	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘(𝑛 − 1)) ∈ ℝ)
26	5	nnred 12229	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 𝑛 ∈ ℝ)
27		peano2rem 11529	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
28	26, 27	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 − 1) ∈ ℝ)
29	26, 28	remulcld 11246	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 · (𝑛 − 1)) ∈ ℝ)
30	29, 22	remulcld 11246	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((𝑛 · (𝑛 − 1)) · ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛)))) ∈ ℝ)
31	5	nncnd 12230	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 𝑛 ∈ ℂ)
32		ax-1cn 11170	. . . . . . . 8 ⊢ 1 ∈ ℂ
33		npcan 11471	. . . . . . . 8 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
34	31, 32, 33	sylancl 586	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((𝑛 − 1) + 1) = 𝑛)
35	34	fveq2d 6895	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
36	15	rpge0d 13022	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 0 ≤ (𝑛 − 1))
37		loglesqrt 26273	. . . . . . 7 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 0 ≤ (𝑛 − 1)) → (log‘((𝑛 − 1) + 1)) ≤ (√‘(𝑛 − 1)))
38	28, 36, 37	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (log‘((𝑛 − 1) + 1)) ≤ (√‘(𝑛 − 1)))
39	35, 38	eqbrtrrd 5172	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (log‘𝑛) ≤ (√‘(𝑛 − 1)))
40	19	rpred 13018	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘𝑛) ∈ ℝ)
41	40, 25	readdcld 11245	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) + (√‘(𝑛 − 1))) ∈ ℝ)
42		remulcl 11197	. . . . . . . . . . 11 ⊢ (((√‘𝑛) ∈ ℝ ∧ 2 ∈ ℝ) → ((√‘𝑛) · 2) ∈ ℝ)
43	40, 14, 42	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) · 2) ∈ ℝ)
44	40, 25	resubcld 11644	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) − (√‘(𝑛 − 1))) ∈ ℝ)
45	26	lem1d 12149	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 − 1) ≤ 𝑛)
46	6	rpge0d 13022	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 0 ≤ 𝑛)
47	28, 36, 26, 46	sqrtled 15375	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((𝑛 − 1) ≤ 𝑛 ↔ (√‘(𝑛 − 1)) ≤ (√‘𝑛)))
48	45, 47	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘(𝑛 − 1)) ≤ (√‘𝑛))
49	40, 25	subge0d 11806	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (0 ≤ ((√‘𝑛) − (√‘(𝑛 − 1))) ↔ (√‘(𝑛 − 1)) ≤ (√‘𝑛)))
50	48, 49	mpbird 256	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 0 ≤ ((√‘𝑛) − (√‘(𝑛 − 1))))
51	25, 40, 40, 48	leadd2dd 11831	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) + (√‘(𝑛 − 1))) ≤ ((√‘𝑛) + (√‘𝑛)))
52	19	rpcnd 13020	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘𝑛) ∈ ℂ)
53	52	times2d 12458	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) · 2) = ((√‘𝑛) + (√‘𝑛)))
54	51, 53	breqtrrd 5176	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) + (√‘(𝑛 − 1))) ≤ ((√‘𝑛) · 2))
55	41, 43, 44, 50, 54	lemul1ad 12155	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛) + (√‘(𝑛 − 1))) · ((√‘𝑛) − (√‘(𝑛 − 1)))) ≤ (((√‘𝑛) · 2) · ((√‘𝑛) − (√‘(𝑛 − 1)))))
56	31	sqsqrtd 15388	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛)↑2) = 𝑛)
57		subcl 11461	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − 1) ∈ ℂ)
58	31, 32, 57	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 − 1) ∈ ℂ)
59	58	sqsqrtd 15388	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘(𝑛 − 1))↑2) = (𝑛 − 1))
60	56, 59	oveq12d 7429	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛)↑2) − ((√‘(𝑛 − 1))↑2)) = (𝑛 − (𝑛 − 1)))
61	16	rpcnd 13020	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘(𝑛 − 1)) ∈ ℂ)
62		subsq 14176	. . . . . . . . . . 11 ⊢ (((√‘𝑛) ∈ ℂ ∧ (√‘(𝑛 − 1)) ∈ ℂ) → (((√‘𝑛)↑2) − ((√‘(𝑛 − 1))↑2)) = (((√‘𝑛) + (√‘(𝑛 − 1))) · ((√‘𝑛) − (√‘(𝑛 − 1)))))
63	52, 61, 62	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛)↑2) − ((√‘(𝑛 − 1))↑2)) = (((√‘𝑛) + (√‘(𝑛 − 1))) · ((√‘𝑛) − (√‘(𝑛 − 1)))))
64		nncan 11491	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − (𝑛 − 1)) = 1)
65	31, 32, 64	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 − (𝑛 − 1)) = 1)
66	60, 63, 65	3eqtr3d 2780	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛) + (√‘(𝑛 − 1))) · ((√‘𝑛) − (√‘(𝑛 − 1)))) = 1)
67		2cn 12289	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
68	67	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 2 ∈ ℂ)
69	44	recnd 11244	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) − (√‘(𝑛 − 1))) ∈ ℂ)
70	52, 68, 69	mulassd 11239	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛) · 2) · ((√‘𝑛) − (√‘(𝑛 − 1)))) = ((√‘𝑛) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))))
71	55, 66, 70	3brtr3d 5179	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 1 ≤ ((√‘𝑛) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))))
72		1red 11217	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 1 ∈ ℝ)
73		remulcl 11197	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ ((√‘𝑛) − (√‘(𝑛 − 1))) ∈ ℝ) → (2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) ∈ ℝ)
74	14, 44, 73	sylancr 587	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) ∈ ℝ)
75	40, 74	remulcld 11246	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))) ∈ ℝ)
76	72, 75, 16	lemul1d 13061	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (1 ≤ ((√‘𝑛) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))) ↔ (1 · (√‘(𝑛 − 1))) ≤ (((√‘𝑛) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))) · (√‘(𝑛 − 1)))))
77	71, 76	mpbid 231	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (1 · (√‘(𝑛 − 1))) ≤ (((√‘𝑛) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))) · (√‘(𝑛 − 1))))
78	61	mullidd 11234	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (1 · (√‘(𝑛 − 1))) = (√‘(𝑛 − 1)))
79	74	recnd 11244	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) ∈ ℂ)
80	52, 79, 61	mul32d 11426	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))) · (√‘(𝑛 − 1))) = (((√‘𝑛) · (√‘(𝑛 − 1))) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))))
81	77, 78, 80	3brtr3d 5179	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘(𝑛 − 1)) ≤ (((√‘𝑛) · (√‘(𝑛 − 1))) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))))
82		remsqsqrt 15205	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → ((√‘𝑛) · (√‘𝑛)) = 𝑛)
83	26, 46, 82	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) · (√‘𝑛)) = 𝑛)
84		remsqsqrt 15205	. . . . . . . . . . 11 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 0 ≤ (𝑛 − 1)) → ((√‘(𝑛 − 1)) · (√‘(𝑛 − 1))) = (𝑛 − 1))
85	28, 36, 84	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘(𝑛 − 1)) · (√‘(𝑛 − 1))) = (𝑛 − 1))
86	83, 85	oveq12d 7429	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛) · (√‘𝑛)) · ((√‘(𝑛 − 1)) · (√‘(𝑛 − 1)))) = (𝑛 · (𝑛 − 1)))
87	52, 52, 61, 61	mul4d 11428	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛) · (√‘𝑛)) · ((√‘(𝑛 − 1)) · (√‘(𝑛 − 1)))) = (((√‘𝑛) · (√‘(𝑛 − 1))) · ((√‘𝑛) · (√‘(𝑛 − 1)))))
88	86, 87	eqtr3d 2774	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (𝑛 · (𝑛 − 1)) = (((√‘𝑛) · (√‘(𝑛 − 1))) · ((√‘𝑛) · (√‘(𝑛 − 1)))))
89	16	rpcnne0d 13027	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘(𝑛 − 1)) ∈ ℂ ∧ (√‘(𝑛 − 1)) ≠ 0))
90	19	rpcnne0d 13027	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) ∈ ℂ ∧ (√‘𝑛) ≠ 0))
91		divsubdiv 11932	. . . . . . . . . 10 ⊢ (((2 ∈ ℂ ∧ 2 ∈ ℂ) ∧ (((√‘(𝑛 − 1)) ∈ ℂ ∧ (√‘(𝑛 − 1)) ≠ 0) ∧ ((√‘𝑛) ∈ ℂ ∧ (√‘𝑛) ≠ 0))) → ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) = (((2 · (√‘𝑛)) − (2 · (√‘(𝑛 − 1)))) / ((√‘(𝑛 − 1)) · (√‘𝑛))))
92	68, 68, 89, 90, 91	syl22anc 837	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) = (((2 · (√‘𝑛)) − (2 · (√‘(𝑛 − 1)))) / ((√‘(𝑛 − 1)) · (√‘𝑛))))
93	68, 52, 61	subdid 11672	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) = ((2 · (√‘𝑛)) − (2 · (√‘(𝑛 − 1)))))
94	52, 61	mulcomd 11237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) · (√‘(𝑛 − 1))) = ((√‘(𝑛 − 1)) · (√‘𝑛)))
95	93, 94	oveq12d 7429	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1)))) = (((2 · (√‘𝑛)) − (2 · (√‘(𝑛 − 1)))) / ((√‘(𝑛 − 1)) · (√‘𝑛))))
96	92, 95	eqtr4d 2775	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) = ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1)))))
97	88, 96	oveq12d 7429	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((𝑛 · (𝑛 − 1)) · ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛)))) = ((((√‘𝑛) · (√‘(𝑛 − 1))) · ((√‘𝑛) · (√‘(𝑛 − 1)))) · ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1))))))
98	52, 61	mulcld 11236	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) · (√‘(𝑛 − 1))) ∈ ℂ)
99	19, 16	rpmulcld 13034	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) · (√‘(𝑛 − 1))) ∈ ℝ⁺)
100	74, 99	rerpdivcld 13049	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1)))) ∈ ℝ)
101	100	recnd 11244	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1)))) ∈ ℂ)
102	98, 98, 101	mulassd 11239	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((((√‘𝑛) · (√‘(𝑛 − 1))) · ((√‘𝑛) · (√‘(𝑛 − 1)))) · ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1))))) = (((√‘𝑛) · (√‘(𝑛 − 1))) · (((√‘𝑛) · (√‘(𝑛 − 1))) · ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1)))))))
103	99	rpne0d 13023	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((√‘𝑛) · (√‘(𝑛 − 1))) ≠ 0)
104	79, 98, 103	divcan2d 11994	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛) · (√‘(𝑛 − 1))) · ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1))))) = (2 · ((√‘𝑛) − (√‘(𝑛 − 1)))))
105	104	oveq2d 7427	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((√‘𝑛) · (√‘(𝑛 − 1))) · (((√‘𝑛) · (√‘(𝑛 − 1))) · ((2 · ((√‘𝑛) − (√‘(𝑛 − 1)))) / ((√‘𝑛) · (√‘(𝑛 − 1)))))) = (((√‘𝑛) · (√‘(𝑛 − 1))) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))))
106	97, 102, 105	3eqtrd 2776	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((𝑛 · (𝑛 − 1)) · ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛)))) = (((√‘𝑛) · (√‘(𝑛 − 1))) · (2 · ((√‘𝑛) − (√‘(𝑛 − 1))))))
107	81, 106	breqtrrd 5176	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘(𝑛 − 1)) ≤ ((𝑛 · (𝑛 − 1)) · ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛)))))
108	7, 25, 30, 39, 107	letrd 11373	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (log‘𝑛) ≤ ((𝑛 · (𝑛 − 1)) · ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛)))))
109	11	nngt0d 12263	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → 0 < (𝑛 · (𝑛 − 1)))
110		ledivmul 12092	. . . . 5 ⊢ (((log‘𝑛) ∈ ℝ ∧ ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) ∈ ℝ ∧ ((𝑛 · (𝑛 − 1)) ∈ ℝ ∧ 0 < (𝑛 · (𝑛 − 1)))) → (((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) ↔ (log‘𝑛) ≤ ((𝑛 · (𝑛 − 1)) · ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))))))
111	7, 22, 29, 109, 110	syl112anc 1374	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) ↔ (log‘𝑛) ≤ ((𝑛 · (𝑛 − 1)) · ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))))))
112	108, 111	mpbird 256	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))))
113	1, 12, 22, 112	fsumle 15747	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ Σ𝑛 ∈ (2...𝐴)((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))))
114		fvoveq1 7434	. . . . . 6 ⊢ (𝑘 = 𝑛 → (√‘(𝑘 − 1)) = (√‘(𝑛 − 1)))
115	114	oveq2d 7427	. . . . 5 ⊢ (𝑘 = 𝑛 → (2 / (√‘(𝑘 − 1))) = (2 / (√‘(𝑛 − 1))))
116		fvoveq1 7434	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (√‘(𝑘 − 1)) = (√‘((𝑛 + 1) − 1)))
117	116	oveq2d 7427	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (2 / (√‘(𝑘 − 1))) = (2 / (√‘((𝑛 + 1) − 1))))
118		oveq1 7418	. . . . . . . . . 10 ⊢ (𝑘 = 2 → (𝑘 − 1) = (2 − 1))
119		2m1e1 12340	. . . . . . . . . 10 ⊢ (2 − 1) = 1
120	118, 119	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝑘 = 2 → (𝑘 − 1) = 1)
121	120	fveq2d 6895	. . . . . . . 8 ⊢ (𝑘 = 2 → (√‘(𝑘 − 1)) = (√‘1))
122		sqrt1 15220	. . . . . . . 8 ⊢ (√‘1) = 1
123	121, 122	eqtrdi 2788	. . . . . . 7 ⊢ (𝑘 = 2 → (√‘(𝑘 − 1)) = 1)
124	123	oveq2d 7427	. . . . . 6 ⊢ (𝑘 = 2 → (2 / (√‘(𝑘 − 1))) = (2 / 1))
125	67	div1i 11944	. . . . . 6 ⊢ (2 / 1) = 2
126	124, 125	eqtrdi 2788	. . . . 5 ⊢ (𝑘 = 2 → (2 / (√‘(𝑘 − 1))) = 2)
127		fvoveq1 7434	. . . . . 6 ⊢ (𝑘 = (𝐴 + 1) → (√‘(𝑘 − 1)) = (√‘((𝐴 + 1) − 1)))
128	127	oveq2d 7427	. . . . 5 ⊢ (𝑘 = (𝐴 + 1) → (2 / (√‘(𝑘 − 1))) = (2 / (√‘((𝐴 + 1) − 1))))
129		nnz 12581	. . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
130		eluzp1p1 12852	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘1) → (𝐴 + 1) ∈ (ℤ_≥‘(1 + 1)))
131		nnuz 12867	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
132	130, 131	eleq2s 2851	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ (ℤ_≥‘(1 + 1)))
133		df-2 12277	. . . . . . 7 ⊢ 2 = (1 + 1)
134	133	fveq2i 6894	. . . . . 6 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
135	132, 134	eleqtrrdi 2844	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ (ℤ_≥‘2))
136		elfzuz 13499	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (2...(𝐴 + 1)) → 𝑘 ∈ (ℤ_≥‘2))
137		uz2m1nn 12909	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (𝑘 − 1) ∈ ℕ)
138	136, 137	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ (2...(𝐴 + 1)) → (𝑘 − 1) ∈ ℕ)
139	138	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (2...(𝐴 + 1))) → (𝑘 − 1) ∈ ℕ)
140	139	nnrpd 13016	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (2...(𝐴 + 1))) → (𝑘 − 1) ∈ ℝ⁺)
141	140	rpsqrtcld 15360	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (2...(𝐴 + 1))) → (√‘(𝑘 − 1)) ∈ ℝ⁺)
142		rerpdivcl 13006	. . . . . . 7 ⊢ ((2 ∈ ℝ ∧ (√‘(𝑘 − 1)) ∈ ℝ⁺) → (2 / (√‘(𝑘 − 1))) ∈ ℝ)
143	14, 141, 142	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (2...(𝐴 + 1))) → (2 / (√‘(𝑘 − 1))) ∈ ℝ)
144	143	recnd 11244	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (2...(𝐴 + 1))) → (2 / (√‘(𝑘 − 1))) ∈ ℂ)
145	115, 117, 126, 128, 129, 135, 144	telfsum 15752	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((2 / (√‘(𝑛 − 1))) − (2 / (√‘((𝑛 + 1) − 1)))) = (2 − (2 / (√‘((𝐴 + 1) − 1)))))
146		pncan 11468	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
147	31, 32, 146	sylancl 586	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((𝑛 + 1) − 1) = 𝑛)
148	147	fveq2d 6895	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (√‘((𝑛 + 1) − 1)) = (√‘𝑛))
149	148	oveq2d 7427	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → (2 / (√‘((𝑛 + 1) − 1))) = (2 / (√‘𝑛)))
150	149	oveq2d 7427	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑛 ∈ (2...𝐴)) → ((2 / (√‘(𝑛 − 1))) − (2 / (√‘((𝑛 + 1) − 1)))) = ((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))))
151	150	sumeq2dv 15651	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((2 / (√‘(𝑛 − 1))) − (2 / (√‘((𝑛 + 1) − 1)))) = Σ𝑛 ∈ (2...𝐴)((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))))
152		nncn 12222	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
153		pncan 11468	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
154	152, 32, 153	sylancl 586	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐴 + 1) − 1) = 𝐴)
155	154	fveq2d 6895	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (√‘((𝐴 + 1) − 1)) = (√‘𝐴))
156	155	oveq2d 7427	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2 / (√‘((𝐴 + 1) − 1))) = (2 / (√‘𝐴)))
157	156	oveq2d 7427	. . . 4 ⊢ (𝐴 ∈ ℕ → (2 − (2 / (√‘((𝐴 + 1) − 1)))) = (2 − (2 / (√‘𝐴))))
158	145, 151, 157	3eqtr3d 2780	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) = (2 − (2 / (√‘𝐴))))
159		2rp 12981	. . . . . 6 ⊢ 2 ∈ ℝ⁺
160		nnrp 12987	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ⁺)
161	160	rpsqrtcld 15360	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (√‘𝐴) ∈ ℝ⁺)
162		rpdivcl 13001	. . . . . 6 ⊢ ((2 ∈ ℝ⁺ ∧ (√‘𝐴) ∈ ℝ⁺) → (2 / (√‘𝐴)) ∈ ℝ⁺)
163	159, 161, 162	sylancr 587	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2 / (√‘𝐴)) ∈ ℝ⁺)
164	163	rpge0d 13022	. . . 4 ⊢ (𝐴 ∈ ℕ → 0 ≤ (2 / (√‘𝐴)))
165	163	rpred 13018	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2 / (√‘𝐴)) ∈ ℝ)
166		subge02 11732	. . . . 5 ⊢ ((2 ∈ ℝ ∧ (2 / (√‘𝐴)) ∈ ℝ) → (0 ≤ (2 / (√‘𝐴)) ↔ (2 − (2 / (√‘𝐴))) ≤ 2))
167	14, 165, 166	sylancr 587	. . . 4 ⊢ (𝐴 ∈ ℕ → (0 ≤ (2 / (√‘𝐴)) ↔ (2 − (2 / (√‘𝐴))) ≤ 2))
168	164, 167	mpbid 231	. . 3 ⊢ (𝐴 ∈ ℕ → (2 − (2 / (√‘𝐴))) ≤ 2)
169	158, 168	eqbrtrd 5170	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((2 / (√‘(𝑛 − 1))) − (2 / (√‘𝑛))) ≤ 2)
170	13, 23, 24, 113, 169	letrd 11373	1 ⊢ (𝐴 ∈ ℕ → Σ𝑛 ∈ (2...𝐴)((log‘𝑛) / (𝑛 · (𝑛 − 1))) ≤ 2)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11250 ≤ cle 11251 − cmin 11446 / cdiv 11873 ℕcn 12214 2c2 12269 ℤ_≥cuz 12824 ℝ⁺crp 12976 ...cfz 13486 ↑cexp 14029 √csqrt 15182 Σcsu 15634 logclog 26070

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 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-ef 16013 df-sin 16015 df-cos 16016 df-tan 16017 df-pi 16018 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-cmp 22898 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-limc 25390 df-dv 25391 df-log 26072 df-cxp 26073

This theorem is referenced by: rplogsumlem2 26995

W3C validator