stirlinglem12 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem12 46513

Description: The sequence 𝐵 is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypotheses**
Ref	Expression
stirlinglem12.1	⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
stirlinglem12.2	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴‘𝑛)))
stirlinglem12.3	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))

**Assertion**
Ref	Expression
stirlinglem12	⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑁))

Distinct variable group: 𝑛,𝑁 Allowed substitution hints: 𝐴(𝑛) 𝐵(𝑛) 𝐹(𝑛)

Proof of Theorem stirlinglem12 Dummy variables 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn 12185	. . . . 5 ⊢ 1 ∈ ℕ
2		stirlinglem12.1	. . . . . . 7 ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
3	2	stirlinglem2 46503	. . . . . 6 ⊢ (1 ∈ ℕ → (𝐴‘1) ∈ ℝ⁺)
4		relogcl 26539	. . . . . 6 ⊢ ((𝐴‘1) ∈ ℝ⁺ → (log‘(𝐴‘1)) ∈ ℝ)
5	1, 3, 4	mp2b 10	. . . . 5 ⊢ (log‘(𝐴‘1)) ∈ ℝ
6		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑛1
7		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑛log
8		nfmpt1 5184	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
9	2, 8	nfcxfr 2896	. . . . . . . 8 ⊢ Ⅎ𝑛𝐴
10	9, 6	nffv 6850	. . . . . . 7 ⊢ Ⅎ𝑛(𝐴‘1)
11	7, 10	nffv 6850	. . . . . 6 ⊢ Ⅎ𝑛(log‘(𝐴‘1))
12		2fveq3 6845	. . . . . 6 ⊢ (𝑛 = 1 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘1)))
13		stirlinglem12.2	. . . . . 6 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴‘𝑛)))
14	6, 11, 12, 13	fvmptf 6969	. . . . 5 ⊢ ((1 ∈ ℕ ∧ (log‘(𝐴‘1)) ∈ ℝ) → (𝐵‘1) = (log‘(𝐴‘1)))
15	1, 5, 14	mp2an 693	. . . 4 ⊢ (𝐵‘1) = (log‘(𝐴‘1))
16	15, 5	eqeltri 2832	. . 3 ⊢ (𝐵‘1) ∈ ℝ
17	16	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘1) ∈ ℝ)
18	2	stirlinglem2 46503	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐴‘𝑁) ∈ ℝ⁺)
19	18	relogcld 26587	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(𝐴‘𝑁)) ∈ ℝ)
20		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑛𝑁
21	9, 20	nffv 6850	. . . . . 6 ⊢ Ⅎ𝑛(𝐴‘𝑁)
22	7, 21	nffv 6850	. . . . 5 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑁))
23		2fveq3 6845	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑁)))
24	20, 22, 23, 13	fvmptf 6969	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (log‘(𝐴‘𝑁)) ∈ ℝ) → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
25	19, 24	mpdan 688	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
26	25, 19	eqeltrd 2836	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) ∈ ℝ)
27		4re 12265	. . . 4 ⊢ 4 ∈ ℝ
28		4ne0 12289	. . . 4 ⊢ 4 ≠ 0
29	27, 28	rereccli 11920	. . 3 ⊢ (1 / 4) ∈ ℝ
30	29	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℝ)
31		fveq2 6840	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
32		fveq2 6840	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝐵‘𝑘) = (𝐵‘(𝑗 + 1)))
33		fveq2 6840	. . . . 5 ⊢ (𝑘 = 1 → (𝐵‘𝑘) = (𝐵‘1))
34		fveq2 6840	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐵‘𝑘) = (𝐵‘𝑁))
35		elnnuz 12828	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
36	35	biimpi 216	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
37		elfznn 13507	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
38	2	stirlinglem2 46503	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ∈ ℝ⁺)
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℝ⁺)
40	39	relogcld 26587	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (log‘(𝐴‘𝑘)) ∈ ℝ)
41		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑘
42	9, 41	nffv 6850	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝐴‘𝑘)
43	7, 42	nffv 6850	. . . . . . . . 9 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑘))
44		2fveq3 6845	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑘)))
45	41, 43, 44, 13	fvmptf 6969	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (log‘(𝐴‘𝑘)) ∈ ℝ) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
46	37, 40, 45	syl2anc 585	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
47	46	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
48	39	rpcnd 12988	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℂ)
49	48	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
50	38	rpne0d 12991	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ≠ 0)
51	37, 50	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ≠ 0)
52	51	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ≠ 0)
53	49, 52	logcld 26534	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘(𝐴‘𝑘)) ∈ ℂ)
54	47, 53	eqeltrd 2836	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℂ)
55	31, 32, 33, 34, 36, 54	telfsumo 15765	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = ((𝐵‘1) − (𝐵‘𝑁)))
56		nnz 12545	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
57		fzoval 13614	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (1..^𝑁) = (1...(𝑁 − 1)))
58	56, 57	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1..^𝑁) = (1...(𝑁 − 1)))
59	58	sumeq1d 15662	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
60	55, 59	eqtr3d 2773	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
61		fzfid 13935	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
62		elfznn 13507	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℕ)
63	62	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → 𝑗 ∈ ℕ)
64	2	stirlinglem2 46503	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝐴‘𝑗) ∈ ℝ⁺)
65	64	relogcld 26587	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘𝑗)) ∈ ℝ)
66		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑗
67	9, 66	nffv 6850	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝐴‘𝑗)
68	7, 67	nffv 6850	. . . . . . . . . 10 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑗))
69		2fveq3 6845	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑗)))
70	66, 68, 69, 13	fvmptf 6969	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ (log‘(𝐴‘𝑗)) ∈ ℝ) → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
71	65, 70	mpdan 688	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
72	71, 65	eqeltrd 2836	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) ∈ ℝ)
73	63, 72	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝐵‘𝑗) ∈ ℝ)
74		peano2nn 12186	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
75	2	stirlinglem2 46503	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
76	74, 75	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
77	76	relogcld 26587	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ)
78		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑗 + 1)
79	9, 78	nffv 6850	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝐴‘(𝑗 + 1))
80	7, 79	nffv 6850	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(log‘(𝐴‘(𝑗 + 1)))
81		2fveq3 6845	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘(𝑗 + 1))))
82	78, 80, 81, 13	fvmptf 6969	. . . . . . . . . 10 ⊢ (((𝑗 + 1) ∈ ℕ ∧ (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ) → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1))))
83	74, 77, 82	syl2anc 585	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1))))
84	83, 77	eqeltrd 2836	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) ∈ ℝ)
85	62, 84	syl 17	. . . . . . 7 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
86	85	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
87	73, 86	resubcld 11578	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ∈ ℝ)
88	61, 87	fsumrecl 15696	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ∈ ℝ)
89	29	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / 4) ∈ ℝ)
90	62	nnred 12189	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℝ)
91		1red 11145	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℝ)
92	90, 91	readdcld 11174	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℝ)
93	90, 92	remulcld 11175	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
94	90	recnd 11173	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℂ)
95		1cnd 11139	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℂ)
96	94, 95	addcld 11164	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℂ)
97	62	nnne0d 12227	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ≠ 0)
98	74	nnne0d 12227	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ≠ 0)
99	62, 98	syl 17	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ≠ 0)
100	94, 96, 97, 99	mulne0d 11802	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ≠ 0)
101	93, 100	rereccld 11982	. . . . . . 7 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
102	101	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
103	89, 102	remulcld 11175	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
104	61, 103	fsumrecl 15696	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
105		eqid 2736	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ ((1 / ((2 · 𝑖) + 1)) · ((1 / ((2 · 𝑗) + 1))↑(2 · 𝑖)))) = (𝑖 ∈ ℕ ↦ ((1 / ((2 · 𝑖) + 1)) · ((1 / ((2 · 𝑗) + 1))↑(2 · 𝑖))))
106		eqid 2736	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖)) = (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖))
107	2, 13, 105, 106	stirlinglem10 46511	. . . . . 6 ⊢ (𝑗 ∈ ℕ → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))))
108	63, 107	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))))
109	61, 87, 103, 108	fsumle 15762	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))))
110	61, 102	fsumrecl 15696	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
111		1red 11145	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
112		4pos 12288	. . . . . . . . 9 ⊢ 0 < 4
113	27, 112	elrpii 12945	. . . . . . . 8 ⊢ 4 ∈ ℝ⁺
114	113	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℝ⁺)
115		0red 11147	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
116		0lt1 11672	. . . . . . . . 9 ⊢ 0 < 1
117	116	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 1)
118	115, 111, 117	ltled 11294	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ 1)
119	111, 114, 118	divge0d 13026	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / 4))
120		eqid 2736	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
121		eluznn 12868	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℕ)
122		stirlinglem12.3	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
123	122	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))))
124		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → 𝑛 = 𝑗)
125	124	oveq1d 7382	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 + 1) = (𝑗 + 1))
126	124, 125	oveq12d 7385	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
127	126	oveq2d 7383	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
128		id 22	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
129		nnre 12181	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
130		1red 11145	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℝ)
131	129, 130	readdcld 11174	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℝ)
132	129, 131	remulcld 11175	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ∈ ℝ)
133		nncn 12182	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
134		1cnd 11139	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℂ)
135	133, 134	addcld 11164	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℂ)
136		nnne0 12211	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ≠ 0)
137	133, 135, 136, 98	mulne0d 11802	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ≠ 0)
138	132, 137	rereccld 11982	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
139	123, 127, 128, 138	fvmptd 6955	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
140	121, 139	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
141	121	nnred 12189	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ)
142		1red 11145	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℝ)
143	141, 142	readdcld 11174	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ)
144	141, 143	remulcld 11175	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
145	141	recnd 11173	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℂ)
146		1cnd 11139	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℂ)
147	145, 146	addcld 11164	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℂ)
148	121	nnne0d 12227	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ≠ 0)
149	121, 98	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ≠ 0)
150	145, 147, 148, 149	mulne0d 11802	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ≠ 0)
151	144, 150	rereccld 11982	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
152		seqeq1 13966	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq𝑁( + , 𝐹) = seq1( + , 𝐹))
153	122	trireciplem 15827	. . . . . . . . . . . . . 14 ⊢ seq1( + , 𝐹) ⇝ 1
154		climrel 15454	. . . . . . . . . . . . . . 15 ⊢ Rel ⇝
155	154	releldmi 5903	. . . . . . . . . . . . . 14 ⊢ (seq1( + , 𝐹) ⇝ 1 → seq1( + , 𝐹) ∈ dom ⇝ )
156	153, 155	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq1( + , 𝐹) ∈ dom ⇝ )
157	152, 156	eqeltrd 2836	. . . . . . . . . . . 12 ⊢ (𝑁 = 1 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
158	157	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
159		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ ℕ)
160		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ¬ 𝑁 = 1)
161		elnn1uz2 12875	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
162	159, 161	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
163	162	ord 865	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (¬ 𝑁 = 1 → 𝑁 ∈ (ℤ_≥‘2)))
164	160, 163	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ (ℤ_≥‘2))
165		uz2m1nn 12873	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
166	164, 165	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 − 1) ∈ ℕ)
167		nncn 12182	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
168	167	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 ∈ ℂ)
169		1cnd 11139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 1 ∈ ℂ)
170	168, 169	npcand 11509	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
171	170	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 = ((𝑁 − 1) + 1))
172	171	seqeq1d 13969	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) = seq((𝑁 − 1) + 1)( + , 𝐹))
173		nnuz 12827	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
174		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℕ)
175	138	recnd 11173	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
176	139, 175	eqeltrd 2836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) ∈ ℂ)
177	176	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 − 1) ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) ∈ ℂ)
178	153	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℕ → seq1( + , 𝐹) ⇝ 1)
179	173, 174, 177, 178	clim2ser 15617	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ ℕ → seq((𝑁 − 1) + 1)( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
180	179	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq((𝑁 − 1) + 1)( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
181	172, 180	eqbrtrd 5107	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
182	154	releldmi 5903	. . . . . . . . . . . . 13 ⊢ (seq𝑁( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
183	181, 182	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
184	159, 166, 183	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
185	158, 184	pm2.61dan 813	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → seq𝑁( + , 𝐹) ∈ dom ⇝ )
186	120, 56, 140, 151, 185	isumrecl 15727	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
187	121	nnrpd 12984	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ⁺)
188	187	rpge0d 12990	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ 𝑗)
189	141, 188	ge0p1rpd 13016	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ⁺)
190	187, 189	rpmulcld 13002	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ∈ ℝ⁺)
191	118	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ 1)
192	142, 190, 191	divge0d 13026	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ (1 / (𝑗 · (𝑗 + 1))))
193	120, 56, 140, 151, 185, 192	isumge0 15728	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))))
194	115, 186, 110, 193	leadd2dd 11765	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + 0) ≤ (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1)))))
195	110	recnd 11173	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
196	195	addridd 11346	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + 0) = Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))))
197	196	eqcomd 2742	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) = (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + 0))
198		id 22	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ)
199	139	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
200	133	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
201		1cnd 11139	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 1 ∈ ℂ)
202	200, 201	addcld 11164	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℂ)
203	200, 202	mulcld 11165	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
204	136	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ≠ 0)
205	98	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ≠ 0)
206	200, 202, 204, 205	mulne0d 11802	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 · (𝑗 + 1)) ≠ 0)
207	203, 206	reccld 11924	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
208	153, 155	mp1i 13	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → seq1( + , 𝐹) ∈ dom ⇝ )
209	173, 120, 198, 199, 207, 208	isumsplit 15805	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1)))))
210	194, 197, 209	3brtr4d 5117	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))))
211		1zzd 12558	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℤ)
212	139	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
213	175	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
214	153	a1i 11	. . . . . . . . 9 ⊢ (⊤ → seq1( + , 𝐹) ⇝ 1)
215	173, 211, 212, 213, 214	isumclim 15719	. . . . . . . 8 ⊢ (⊤ → Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1)
216	215	mptru 1549	. . . . . . 7 ⊢ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1
217	210, 216	breqtrdi 5126	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ 1)
218	110, 111, 30, 119, 217	lemul2ad 12096	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1)))) ≤ ((1 / 4) · 1))
219		4cn 12266	. . . . . . . 8 ⊢ 4 ∈ ℂ
220	219	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℂ)
221	112	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 4)
222	221	gt0ne0d 11714	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ≠ 0)
223	220, 222	reccld 11924	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℂ)
224	102	recnd 11173	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
225	61, 223, 224	fsummulc2 15746	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1)))) = Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))))
226	223	mulridd 11162	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · 1) = (1 / 4))
227	218, 225, 226	3brtr3d 5116	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ≤ (1 / 4))
228	88, 104, 30, 109, 227	letrd 11303	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ (1 / 4))
229	60, 228	eqbrtrd 5107	. 2 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) ≤ (1 / 4))
230	17, 26, 30, 229	subled 11753	1 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2932 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 2c2 12236 4c4 12238 ℤcz 12524 ℤ_≥cuz 12788 ℝ⁺crp 12942 ...cfz 13461 ..^cfzo 13608 seqcseq 13963 ↑cexp 14023 !cfa 14235 √csqrt 15195 ⇝ cli 15446 Σcsu 15648 eceu 16027 logclog 26518

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-e 16033 df-sin 16034 df-cos 16035 df-tan 16036 df-pi 16037 df-dvds 16222 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-ulm 26342 df-log 26520 df-cxp 26521

This theorem is referenced by: stirlinglem13 46514

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