stirlinglem12 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem12 43516

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 11914	. . . . 5 ⊢ 1 ∈ ℕ
2		stirlinglem12.1	. . . . . . 7 ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
3	2	stirlinglem2 43506	. . . . . 6 ⊢ (1 ∈ ℕ → (𝐴‘1) ∈ ℝ⁺)
4		relogcl 25636	. . . . . 6 ⊢ ((𝐴‘1) ∈ ℝ⁺ → (log‘(𝐴‘1)) ∈ ℝ)
5	1, 3, 4	mp2b 10	. . . . 5 ⊢ (log‘(𝐴‘1)) ∈ ℝ
6		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑛1
7		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑛log
8		nfmpt1 5178	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
9	2, 8	nfcxfr 2904	. . . . . . . 8 ⊢ Ⅎ𝑛𝐴
10	9, 6	nffv 6766	. . . . . . 7 ⊢ Ⅎ𝑛(𝐴‘1)
11	7, 10	nffv 6766	. . . . . 6 ⊢ Ⅎ𝑛(log‘(𝐴‘1))
12		2fveq3 6761	. . . . . 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 6878	. . . . 5 ⊢ ((1 ∈ ℕ ∧ (log‘(𝐴‘1)) ∈ ℝ) → (𝐵‘1) = (log‘(𝐴‘1)))
15	1, 5, 14	mp2an 688	. . . 4 ⊢ (𝐵‘1) = (log‘(𝐴‘1))
16	15, 5	eqeltri 2835	. . 3 ⊢ (𝐵‘1) ∈ ℝ
17	16	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘1) ∈ ℝ)
18	2	stirlinglem2 43506	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐴‘𝑁) ∈ ℝ⁺)
19	18	relogcld 25683	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(𝐴‘𝑁)) ∈ ℝ)
20		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑛𝑁
21	9, 20	nffv 6766	. . . . . 6 ⊢ Ⅎ𝑛(𝐴‘𝑁)
22	7, 21	nffv 6766	. . . . 5 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑁))
23		2fveq3 6761	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑁)))
24	20, 22, 23, 13	fvmptf 6878	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (log‘(𝐴‘𝑁)) ∈ ℝ) → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
25	19, 24	mpdan 683	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
26	25, 19	eqeltrd 2839	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) ∈ ℝ)
27		4re 11987	. . . 4 ⊢ 4 ∈ ℝ
28		4ne0 12011	. . . 4 ⊢ 4 ≠ 0
29	27, 28	rereccli 11670	. . 3 ⊢ (1 / 4) ∈ ℝ
30	29	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℝ)
31		fveq2 6756	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
32		fveq2 6756	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝐵‘𝑘) = (𝐵‘(𝑗 + 1)))
33		fveq2 6756	. . . . 5 ⊢ (𝑘 = 1 → (𝐵‘𝑘) = (𝐵‘1))
34		fveq2 6756	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐵‘𝑘) = (𝐵‘𝑁))
35		elnnuz 12551	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
36	35	biimpi 215	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
37		elfznn 13214	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
38	2	stirlinglem2 43506	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ∈ ℝ⁺)
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℝ⁺)
40	39	relogcld 25683	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (log‘(𝐴‘𝑘)) ∈ ℝ)
41		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑘
42	9, 41	nffv 6766	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝐴‘𝑘)
43	7, 42	nffv 6766	. . . . . . . . 9 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑘))
44		2fveq3 6761	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑘)))
45	41, 43, 44, 13	fvmptf 6878	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (log‘(𝐴‘𝑘)) ∈ ℝ) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
46	37, 40, 45	syl2anc 583	. . . . . . 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 12703	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℂ)
49	48	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
50	38	rpne0d 12706	. . . . . . . . 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 25631	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘(𝐴‘𝑘)) ∈ ℂ)
54	47, 53	eqeltrd 2839	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℂ)
55	31, 32, 33, 34, 36, 54	telfsumo 15442	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = ((𝐵‘1) − (𝐵‘𝑁)))
56		nnz 12272	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
57		fzoval 13317	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (1..^𝑁) = (1...(𝑁 − 1)))
58	56, 57	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1..^𝑁) = (1...(𝑁 − 1)))
59	58	sumeq1d 15341	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
60	55, 59	eqtr3d 2780	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
61		fzfid 13621	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
62		elfznn 13214	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℕ)
63	62	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → 𝑗 ∈ ℕ)
64	2	stirlinglem2 43506	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝐴‘𝑗) ∈ ℝ⁺)
65	64	relogcld 25683	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘𝑗)) ∈ ℝ)
66		nfcv 2906	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑗
67	9, 66	nffv 6766	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝐴‘𝑗)
68	7, 67	nffv 6766	. . . . . . . . . 10 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑗))
69		2fveq3 6761	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑗)))
70	66, 68, 69, 13	fvmptf 6878	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ (log‘(𝐴‘𝑗)) ∈ ℝ) → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
71	65, 70	mpdan 683	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
72	71, 65	eqeltrd 2839	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) ∈ ℝ)
73	63, 72	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝐵‘𝑗) ∈ ℝ)
74		peano2nn 11915	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
75	2	stirlinglem2 43506	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
76	74, 75	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
77	76	relogcld 25683	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ)
78		nfcv 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑗 + 1)
79	9, 78	nffv 6766	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝐴‘(𝑗 + 1))
80	7, 79	nffv 6766	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(log‘(𝐴‘(𝑗 + 1)))
81		2fveq3 6761	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘(𝑗 + 1))))
82	78, 80, 81, 13	fvmptf 6878	. . . . . . . . . 10 ⊢ (((𝑗 + 1) ∈ ℕ ∧ (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ) → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1))))
83	74, 77, 82	syl2anc 583	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1))))
84	83, 77	eqeltrd 2839	. . . . . . . 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 11333	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ∈ ℝ)
88	61, 87	fsumrecl 15374	. . . 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 11918	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℝ)
91		1red 10907	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℝ)
92	90, 91	readdcld 10935	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℝ)
93	90, 92	remulcld 10936	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
94	90	recnd 10934	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℂ)
95		1cnd 10901	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℂ)
96	94, 95	addcld 10925	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℂ)
97	62	nnne0d 11953	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ≠ 0)
98	74	nnne0d 11953	. . . . . . . . . 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 11557	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ≠ 0)
101	93, 100	rereccld 11732	. . . . . . 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 10936	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
104	61, 103	fsumrecl 15374	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
105		eqid 2738	. . . . . . 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 2738	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖)) = (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖))
107	2, 13, 105, 106	stirlinglem10 43514	. . . . . 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 15439	. . . 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 15374	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
111		1red 10907	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
112		4pos 12010	. . . . . . . . 9 ⊢ 0 < 4
113	27, 112	elrpii 12662	. . . . . . . 8 ⊢ 4 ∈ ℝ⁺
114	113	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℝ⁺)
115		0red 10909	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
116		0lt1 11427	. . . . . . . . 9 ⊢ 0 < 1
117	116	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 1)
118	115, 111, 117	ltled 11053	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ 1)
119	111, 114, 118	divge0d 12741	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / 4))
120		eqid 2738	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
121		eluznn 12587	. . . . . . . . . . 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 7270	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 + 1) = (𝑗 + 1))
126	124, 125	oveq12d 7273	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
127	126	oveq2d 7271	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
128		id 22	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
129		nnre 11910	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
130		1red 10907	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℝ)
131	129, 130	readdcld 10935	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℝ)
132	129, 131	remulcld 10936	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ∈ ℝ)
133		nncn 11911	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
134		1cnd 10901	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℂ)
135	133, 134	addcld 10925	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℂ)
136		nnne0 11937	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ≠ 0)
137	133, 135, 136, 98	mulne0d 11557	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ≠ 0)
138	132, 137	rereccld 11732	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
139	123, 127, 128, 138	fvmptd 6864	. . . . . . . . . . 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 11918	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ)
142		1red 10907	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℝ)
143	141, 142	readdcld 10935	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ)
144	141, 143	remulcld 10936	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
145	141	recnd 10934	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℂ)
146		1cnd 10901	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℂ)
147	145, 146	addcld 10925	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℂ)
148	121	nnne0d 11953	. . . . . . . . . . . 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 11557	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ≠ 0)
151	144, 150	rereccld 11732	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
152		seqeq1 13652	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq𝑁( + , 𝐹) = seq1( + , 𝐹))
153	122	trireciplem 15502	. . . . . . . . . . . . . 14 ⊢ seq1( + , 𝐹) ⇝ 1
154		climrel 15129	. . . . . . . . . . . . . . 15 ⊢ Rel ⇝
155	154	releldmi 5846	. . . . . . . . . . . . . 14 ⊢ (seq1( + , 𝐹) ⇝ 1 → seq1( + , 𝐹) ∈ dom ⇝ )
156	153, 155	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq1( + , 𝐹) ∈ dom ⇝ )
157	152, 156	eqeltrd 2839	. . . . . . . . . . . 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 12594	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
162	159, 161	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
163	162	ord 860	. . . . . . . . . . . . . 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 12592	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
166	164, 165	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 − 1) ∈ ℕ)
167		nncn 11911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
168	167	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 ∈ ℂ)
169		1cnd 10901	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 1 ∈ ℂ)
170	168, 169	npcand 11266	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
171	170	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 = ((𝑁 − 1) + 1))
172	171	seqeq1d 13655	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) = seq((𝑁 − 1) + 1)( + , 𝐹))
173		nnuz 12550	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
174		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℕ)
175	138	recnd 10934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
176	139, 175	eqeltrd 2839	. . . . . . . . . . . . . . . . 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 15294	. . . . . . . . . . . . . . 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 5092	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
182	154	releldmi 5846	. . . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
185	158, 184	pm2.61dan 809	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → seq𝑁( + , 𝐹) ∈ dom ⇝ )
186	120, 56, 140, 151, 185	isumrecl 15405	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
187	121	nnrpd 12699	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ⁺)
188	187	rpge0d 12705	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ 𝑗)
189	141, 188	ge0p1rpd 12731	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ⁺)
190	187, 189	rpmulcld 12717	. . . . . . . . . . 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 12741	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ (1 / (𝑗 · (𝑗 + 1))))
193	120, 56, 140, 151, 185, 192	isumge0 15406	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))))
194	115, 186, 110, 193	leadd2dd 11520	. . . . . . . 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 10934	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
196	195	addid1d 11105	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + 0) = Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))))
197	196	eqcomd 2744	. . . . . . . 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 10901	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 1 ∈ ℂ)
202	200, 201	addcld 10925	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℂ)
203	200, 202	mulcld 10926	. . . . . . . . . 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 11557	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 · (𝑗 + 1)) ≠ 0)
207	203, 206	reccld 11674	. . . . . . . . 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 15480	. . . . . . . 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 5102	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))))
211		1zzd 12281	. . . . . . . . 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 15397	. . . . . . . 8 ⊢ (⊤ → Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1)
216	215	mptru 1546	. . . . . . 7 ⊢ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1
217	210, 216	breqtrdi 5111	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ 1)
218	110, 111, 30, 119, 217	lemul2ad 11845	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1)))) ≤ ((1 / 4) · 1))
219		4cn 11988	. . . . . . . 8 ⊢ 4 ∈ ℂ
220	219	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℂ)
221	112	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 4)
222	221	gt0ne0d 11469	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ≠ 0)
223	220, 222	reccld 11674	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℂ)
224	102	recnd 10934	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
225	61, 223, 224	fsummulc2 15424	. . . . 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	mulid1d 10923	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · 1) = (1 / 4))
227	218, 225, 226	3brtr3d 5101	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ≤ (1 / 4))
228	88, 104, 30, 109, 227	letrd 11062	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ (1 / 4))
229	60, 228	eqbrtrd 5092	. 2 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) ≤ (1 / 4))
230	17, 26, 30, 229	subled 11508	1 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 843 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 4c4 11960 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 ...cfz 13168 ..^cfzo 13311 seqcseq 13649 ↑cexp 13710 !cfa 13915 √csqrt 14872 ⇝ cli 15121 Σcsu 15325 eceu 15700 logclog 25615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-e 15706 df-sin 15707 df-cos 15708 df-tan 15709 df-pi 15710 df-dvds 15892 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-ulm 25441 df-log 25617 df-cxp 25618

This theorem is referenced by: stirlinglem13 43517

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