stirlinglem12 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem12 46819

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 12239	. . . . 5 ⊢ 1 ∈ ℕ
2		stirlinglem12.1	. . . . . . 7 ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
3	2	stirlinglem2 46809	. . . . . 6 ⊢ (1 ∈ ℕ → (𝐴‘1) ∈ ℝ⁺)
4		relogcl 26740	. . . . . 6 ⊢ ((𝐴‘1) ∈ ℝ⁺ → (log‘(𝐴‘1)) ∈ ℝ)
5	1, 3, 4	mp2b 10	. . . . 5 ⊢ (log‘(𝐴‘1)) ∈ ℝ
6		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑛1
7		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑛log
8		nfmpt1 5210	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
9	2, 8	nfcxfr 2923	. . . . . . . 8 ⊢ Ⅎ𝑛𝐴
10	9, 6	nffv 6891	. . . . . . 7 ⊢ Ⅎ𝑛(𝐴‘1)
11	7, 10	nffv 6891	. . . . . 6 ⊢ Ⅎ𝑛(log‘(𝐴‘1))
12		2fveq3 6886	. . . . . 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 7011	. . . . 5 ⊢ ((1 ∈ ℕ ∧ (log‘(𝐴‘1)) ∈ ℝ) → (𝐵‘1) = (log‘(𝐴‘1)))
15	1, 5, 14	mp2an 704	. . . 4 ⊢ (𝐵‘1) = (log‘(𝐴‘1))
16	15, 5	eqeltri 2859	. . 3 ⊢ (𝐵‘1) ∈ ℝ
17	16	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘1) ∈ ℝ)
18	2	stirlinglem2 46809	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐴‘𝑁) ∈ ℝ⁺)
19	18	relogcld 26788	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(𝐴‘𝑁)) ∈ ℝ)
20		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑛𝑁
21	9, 20	nffv 6891	. . . . . 6 ⊢ Ⅎ𝑛(𝐴‘𝑁)
22	7, 21	nffv 6891	. . . . 5 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑁))
23		2fveq3 6886	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑁)))
24	20, 22, 23, 13	fvmptf 7011	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (log‘(𝐴‘𝑁)) ∈ ℝ) → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
25	19, 24	mpdan 699	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
26	25, 19	eqeltrd 2863	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) ∈ ℝ)
27		4re 12320	. . . 4 ⊢ 4 ∈ ℝ
28		4ne0 12347	. . . 4 ⊢ 4 ≠ 0
29	27, 28	rereccli 11975	. . 3 ⊢ (1 / 4) ∈ ℝ
30	29	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℝ)
31		fveq2 6881	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
32		fveq2 6881	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝐵‘𝑘) = (𝐵‘(𝑗 + 1)))
33		fveq2 6881	. . . . 5 ⊢ (𝑘 = 1 → (𝐵‘𝑘) = (𝐵‘1))
34		fveq2 6881	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐵‘𝑘) = (𝐵‘𝑁))
35		elnnuz 12897	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
36	35	biimpi 219	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
37		elfznn 13577	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
38	2	stirlinglem2 46809	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ∈ ℝ⁺)
39	37, 38	syl 18	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℝ⁺)
40	39	relogcld 26788	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (log‘(𝐴‘𝑘)) ∈ ℝ)
41		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑘
42	9, 41	nffv 6891	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝐴‘𝑘)
43	7, 42	nffv 6891	. . . . . . . . 9 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑘))
44		2fveq3 6886	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑘)))
45	41, 43, 44, 13	fvmptf 7011	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (log‘(𝐴‘𝑘)) ∈ ℝ) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
46	37, 40, 45	syl2anc 595	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
47	46	adantl 486	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
48	39	rpcnd 13057	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℂ)
49	48	adantl 486	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
50	38	rpne0d 13060	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ≠ 0)
51	37, 50	syl 18	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ≠ 0)
52	51	adantl 486	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ≠ 0)
53	49, 52	logcld 26735	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘(𝐴‘𝑘)) ∈ ℂ)
54	47, 53	eqeltrd 2863	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℂ)
55	31, 32, 33, 34, 36, 54	telfsumo 15850	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = ((𝐵‘1) − (𝐵‘𝑁)))
56		nnz 12607	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
57		fzoval 13684	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (1..^𝑁) = (1...(𝑁 − 1)))
58	56, 57	syl 18	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1..^𝑁) = (1...(𝑁 − 1)))
59	58	sumeq1d 15747	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
60	55, 59	eqtr3d 2800	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
61		fzfid 14005	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
62		elfznn 13577	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℕ)
63	62	adantl 486	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → 𝑗 ∈ ℕ)
64	2	stirlinglem2 46809	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝐴‘𝑗) ∈ ℝ⁺)
65	64	relogcld 26788	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘𝑗)) ∈ ℝ)
66		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑗
67	9, 66	nffv 6891	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝐴‘𝑗)
68	7, 67	nffv 6891	. . . . . . . . . 10 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑗))
69		2fveq3 6886	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑗)))
70	66, 68, 69, 13	fvmptf 7011	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ (log‘(𝐴‘𝑗)) ∈ ℝ) → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
71	65, 70	mpdan 699	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
72	71, 65	eqeltrd 2863	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) ∈ ℝ)
73	63, 72	syl 18	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝐵‘𝑗) ∈ ℝ)
74		peano2nn 12240	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
75	2	stirlinglem2 46809	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
76	74, 75	syl 18	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
77	76	relogcld 26788	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ)
78		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑗 + 1)
79	9, 78	nffv 6891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝐴‘(𝑗 + 1))
80	7, 79	nffv 6891	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(log‘(𝐴‘(𝑗 + 1)))
81		2fveq3 6886	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘(𝑗 + 1))))
82	78, 80, 81, 13	fvmptf 7011	. . . . . . . . . 10 ⊢ (((𝑗 + 1) ∈ ℕ ∧ (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ) → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1))))
83	74, 77, 82	syl2anc 595	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1))))
84	83, 77	eqeltrd 2863	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) ∈ ℝ)
85	62, 84	syl 18	. . . . . . 7 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
86	85	adantl 486	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
87	73, 86	resubcld 11637	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ∈ ℝ)
88	61, 87	fsumrecl 15781	. . . 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 12243	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℝ)
91		1red 11204	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℝ)
92	90, 91	readdcld 11233	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℝ)
93	90, 92	remulcld 11234	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
94	90	recnd 11232	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℂ)
95		1cnd 11197	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℂ)
96	94, 95	addcld 11223	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℂ)
97	62	nnne0d 12281	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ≠ 0)
98	74	nnne0d 12281	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ≠ 0)
99	62, 98	syl 18	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ≠ 0)
100	94, 96, 97, 99	mulne0d 11861	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ≠ 0)
101	93, 100	rereccld 12037	. . . . . . 7 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
102	101	adantl 486	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
103	89, 102	remulcld 11234	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
104	61, 103	fsumrecl 15781	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
105		eqid 2763	. . . . . . 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 2763	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖)) = (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖))
107	2, 13, 105, 106	stirlinglem10 46817	. . . . . 6 ⊢ (𝑗 ∈ ℕ → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))))
108	63, 107	syl 18	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))))
109	61, 87, 103, 108	fsumle 15847	. . . 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 15781	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
111		1red 11204	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
112		4pos 12346	. . . . . . . . 9 ⊢ 0 < 4
113	27, 112	elrpii 13014	. . . . . . . 8 ⊢ 4 ∈ ℝ⁺
114	113	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℝ⁺)
115		0red 11206	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
116		0lt1 11731	. . . . . . . . 9 ⊢ 0 < 1
117	116	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 1)
118	115, 111, 117	ltled 11353	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ 1)
119	111, 114, 118	divge0d 13095	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / 4))
120		eqid 2763	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
121		eluznn 12937	. . . . . . . . . . 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 489	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → 𝑛 = 𝑗)
125	124	oveq1d 7425	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 + 1) = (𝑗 + 1))
126	124, 125	oveq12d 7428	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
127	126	oveq2d 7426	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
128		id 23	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
129		nnre 12235	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
130		1red 11204	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℝ)
131	129, 130	readdcld 11233	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℝ)
132	129, 131	remulcld 11234	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ∈ ℝ)
133		nncn 12236	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
134		1cnd 11197	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℂ)
135	133, 134	addcld 11223	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℂ)
136		nnne0 12265	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ≠ 0)
137	133, 135, 136, 98	mulne0d 11861	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ≠ 0)
138	132, 137	rereccld 12037	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
139	123, 127, 128, 138	fvmptd 6997	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
140	121, 139	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
141	121	nnred 12243	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ)
142		1red 11204	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℝ)
143	141, 142	readdcld 11233	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ)
144	141, 143	remulcld 11234	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
145	141	recnd 11232	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℂ)
146		1cnd 11197	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℂ)
147	145, 146	addcld 11223	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℂ)
148	121	nnne0d 12281	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ≠ 0)
149	121, 98	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ≠ 0)
150	145, 147, 148, 149	mulne0d 11861	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ≠ 0)
151	144, 150	rereccld 12037	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
152		seqeq1 14036	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq𝑁( + , 𝐹) = seq1( + , 𝐹))
153	122	trireciplem 15912	. . . . . . . . . . . . . 14 ⊢ seq1( + , 𝐹) ⇝ 1
154		climrel 15539	. . . . . . . . . . . . . . 15 ⊢ Rel ⇝
155	154	releldmi 5938	. . . . . . . . . . . . . 14 ⊢ (seq1( + , 𝐹) ⇝ 1 → seq1( + , 𝐹) ∈ dom ⇝ )
156	153, 155	mp1i 14	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq1( + , 𝐹) ∈ dom ⇝ )
157	152, 156	eqeltrd 2863	. . . . . . . . . . . 12 ⊢ (𝑁 = 1 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
158	157	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
159		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ ℕ)
160		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ¬ 𝑁 = 1)
161		elnn1uz2 12944	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
162	161	birani 508	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
163	162	ord 877	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (¬ 𝑁 = 1 → 𝑁 ∈ (ℤ_≥‘2)))
164	160, 163	mpd 16	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ (ℤ_≥‘2))
165		uz2m1nn 12942	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
166	164, 165	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 − 1) ∈ ℕ)
167		nncn 12236	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
168	167	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 ∈ ℂ)
169		1cnd 11197	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 1 ∈ ℂ)
170	168, 169	npcand 11568	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
171	170	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 = ((𝑁 − 1) + 1))
172	171	seqeq1d 14039	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) = seq((𝑁 − 1) + 1)( + , 𝐹))
173		nnuz 12896	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
174		id 23	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℕ)
175	138	recnd 11232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
176	139, 175	eqeltrd 2863	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) ∈ ℂ)
177	176	adantl 486	. . . . . . . . . . . . . . . 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 15702	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ ℕ → seq((𝑁 − 1) + 1)( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
180	179	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq((𝑁 − 1) + 1)( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
181	172, 180	eqbrtrd 5133	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
182	154	releldmi 5938	. . . . . . . . . . . . 13 ⊢ (seq𝑁( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
183	181, 182	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
184	159, 166, 183	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
185	158, 184	pm2.61dan 824	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → seq𝑁( + , 𝐹) ∈ dom ⇝ )
186	120, 56, 140, 151, 185	isumrecl 15812	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
187	121	nnrpd 13053	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ⁺)
188	187	rpge0d 13059	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ 𝑗)
189	141, 188	ge0p1rpd 13085	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ⁺)
190	187, 189	rpmulcld 13071	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ∈ ℝ⁺)
191	118	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ 1)
192	142, 190, 191	divge0d 13095	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ (1 / (𝑗 · (𝑗 + 1))))
193	120, 56, 140, 151, 185, 192	isumge0 15813	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))))
194	115, 186, 110, 193	leadd2dd 11824	. . . . . . . 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 11232	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
196	195	addridd 11405	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + 0) = Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))))
197	196	eqcomd 2769	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) = (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + 0))
198		id 23	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ)
199	139	adantl 486	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
200	133	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
201		1cnd 11197	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 1 ∈ ℂ)
202	200, 201	addcld 11223	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℂ)
203	200, 202	mulcld 11224	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
204	136	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ≠ 0)
205	98	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ≠ 0)
206	200, 202, 204, 205	mulne0d 11861	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 · (𝑗 + 1)) ≠ 0)
207	203, 206	reccld 11979	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
208	153, 155	mp1i 14	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → seq1( + , 𝐹) ∈ dom ⇝ )
209	173, 120, 198, 199, 207, 208	isumsplit 15890	. . . . . . . 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 5143	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))))
211		1zzd 12620	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℤ)
212	139	adantl 486	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
213	175	adantl 486	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
214	153	a1i 11	. . . . . . . . 9 ⊢ (⊤ → seq1( + , 𝐹) ⇝ 1)
215	173, 211, 212, 213, 214	isumclim 15804	. . . . . . . 8 ⊢ (⊤ → Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1)
216	215	mptru 1577	. . . . . . 7 ⊢ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1
217	210, 216	breqtrdi 5152	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ 1)
218	110, 111, 30, 119, 217	lemul2ad 12150	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1)))) ≤ ((1 / 4) · 1))
219		4cn 12321	. . . . . . . 8 ⊢ 4 ∈ ℂ
220	219	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℂ)
221	112	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 4)
222	221	gt0ne0d 11773	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ≠ 0)
223	220, 222	reccld 11979	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℂ)
224	102	recnd 11232	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
225	61, 223, 224	fsummulc2 15831	. . . . 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 11221	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · 1) = (1 / 4))
227	218, 225, 226	3brtr3d 5142	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ≤ (1 / 4))
228	88, 104, 30, 109, 227	letrd 11362	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ (1 / 4))
229	60, 228	eqbrtrd 5133	. 2 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) ≤ (1 / 4))
230	17, 26, 30, 229	subled 11812	1 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∨ wo 860 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5661 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 < clt 11238 ≤ cle 11239 − cmin 11436 / cdiv 11866 ℕcn 12228 2c2 12290 4c4 12292 ℤcz 12586 ℤ_≥cuz 12857 ℝ⁺crp 13011 ...cfz 13530 ..^cfzo 13678 seqcseq 14033 ↑cexp 14093 !cfa 14305 √csqrt 15280 ⇝ cli 15531 Σcsu 15733 eceu 16111 logclog 26719

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16116 df-e 16117 df-sin 16118 df-cos 16119 df-tan 16120 df-pi 16121 df-dvds 16306 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-haus 23472 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026 df-ulm 26540 df-log 26721 df-cxp 26722

This theorem is referenced by: stirlinglem13 46820

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