stirlinglem12 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem12 45252

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 12219	. . . . 5 ⊢ 1 ∈ ℕ
2		stirlinglem12.1	. . . . . . 7 ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
3	2	stirlinglem2 45242	. . . . . 6 ⊢ (1 ∈ ℕ → (𝐴‘1) ∈ ℝ⁺)
4		relogcl 26425	. . . . . 6 ⊢ ((𝐴‘1) ∈ ℝ⁺ → (log‘(𝐴‘1)) ∈ ℝ)
5	1, 3, 4	mp2b 10	. . . . 5 ⊢ (log‘(𝐴‘1)) ∈ ℝ
6		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑛1
7		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑛log
8		nfmpt1 5246	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
9	2, 8	nfcxfr 2893	. . . . . . . 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 7009	. . . . 5 ⊢ ((1 ∈ ℕ ∧ (log‘(𝐴‘1)) ∈ ℝ) → (𝐵‘1) = (log‘(𝐴‘1)))
15	1, 5, 14	mp2an 689	. . . 4 ⊢ (𝐵‘1) = (log‘(𝐴‘1))
16	15, 5	eqeltri 2821	. . 3 ⊢ (𝐵‘1) ∈ ℝ
17	16	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘1) ∈ ℝ)
18	2	stirlinglem2 45242	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐴‘𝑁) ∈ ℝ⁺)
19	18	relogcld 26472	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(𝐴‘𝑁)) ∈ ℝ)
20		nfcv 2895	. . . . 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 7009	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (log‘(𝐴‘𝑁)) ∈ ℝ) → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
25	19, 24	mpdan 684	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
26	25, 19	eqeltrd 2825	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) ∈ ℝ)
27		4re 12292	. . . 4 ⊢ 4 ∈ ℝ
28		4ne0 12316	. . . 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 12862	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
36	35	biimpi 215	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
37		elfznn 13526	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
38	2	stirlinglem2 45242	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ∈ ℝ⁺)
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℝ⁺)
40	39	relogcld 26472	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (log‘(𝐴‘𝑘)) ∈ ℝ)
41		nfcv 2895	. . . . . . . . 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 7009	. . . . . . . 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 13014	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℂ)
49	48	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
50	38	rpne0d 13017	. . . . . . . . 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 26420	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘(𝐴‘𝑘)) ∈ ℂ)
54	47, 53	eqeltrd 2825	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℂ)
55	31, 32, 33, 34, 36, 54	telfsumo 15744	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = ((𝐵‘1) − (𝐵‘𝑁)))
56		nnz 12575	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
57		fzoval 13629	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (1..^𝑁) = (1...(𝑁 − 1)))
58	56, 57	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1..^𝑁) = (1...(𝑁 − 1)))
59	58	sumeq1d 15643	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
60	55, 59	eqtr3d 2766	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
61		fzfid 13934	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
62		elfznn 13526	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℕ)
63	62	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → 𝑗 ∈ ℕ)
64	2	stirlinglem2 45242	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝐴‘𝑗) ∈ ℝ⁺)
65	64	relogcld 26472	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘𝑗)) ∈ ℝ)
66		nfcv 2895	. . . . . . . . . 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 7009	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ (log‘(𝐴‘𝑗)) ∈ ℝ) → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
71	65, 70	mpdan 684	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
72	71, 65	eqeltrd 2825	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) ∈ ℝ)
73	63, 72	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝐵‘𝑗) ∈ ℝ)
74		peano2nn 12220	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
75	2	stirlinglem2 45242	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
76	74, 75	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
77	76	relogcld 26472	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ)
78		nfcv 2895	. . . . . . . . . . 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 7009	. . . . . . . . . 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 2825	. . . . . . . 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 11638	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ∈ ℝ)
88	61, 87	fsumrecl 15676	. . . 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 12223	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℝ)
91		1red 11211	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℝ)
92	90, 91	readdcld 11239	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℝ)
93	90, 92	remulcld 11240	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
94	90	recnd 11238	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℂ)
95		1cnd 11205	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℂ)
96	94, 95	addcld 11229	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℂ)
97	62	nnne0d 12258	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ≠ 0)
98	74	nnne0d 12258	. . . . . . . . . 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 11862	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
103	89, 102	remulcld 11240	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
104	61, 103	fsumrecl 15676	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
105		eqid 2724	. . . . . . 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 2724	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖)) = (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖))
107	2, 13, 105, 106	stirlinglem10 45250	. . . . . 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 15741	. . . 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 15676	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
111		1red 11211	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
112		4pos 12315	. . . . . . . . 9 ⊢ 0 < 4
113	27, 112	elrpii 12973	. . . . . . . 8 ⊢ 4 ∈ ℝ⁺
114	113	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℝ⁺)
115		0red 11213	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
116		0lt1 11732	. . . . . . . . 9 ⊢ 0 < 1
117	116	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 1)
118	115, 111, 117	ltled 11358	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ 1)
119	111, 114, 118	divge0d 13052	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / 4))
120		eqid 2724	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
121		eluznn 12898	. . . . . . . . . . 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 7416	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 + 1) = (𝑗 + 1))
126	124, 125	oveq12d 7419	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
127	126	oveq2d 7417	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
128		id 22	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
129		nnre 12215	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
130		1red 11211	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℝ)
131	129, 130	readdcld 11239	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℝ)
132	129, 131	remulcld 11240	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ∈ ℝ)
133		nncn 12216	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
134		1cnd 11205	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℂ)
135	133, 134	addcld 11229	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℂ)
136		nnne0 12242	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ≠ 0)
137	133, 135, 136, 98	mulne0d 11862	. . . . . . . . . . . . 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 6995	. . . . . . . . . . 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 12223	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ)
142		1red 11211	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℝ)
143	141, 142	readdcld 11239	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ)
144	141, 143	remulcld 11240	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
145	141	recnd 11238	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℂ)
146		1cnd 11205	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℂ)
147	145, 146	addcld 11229	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℂ)
148	121	nnne0d 12258	. . . . . . . . . . . 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 11862	. . . . . . . . . . 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 13965	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq𝑁( + , 𝐹) = seq1( + , 𝐹))
153	122	trireciplem 15804	. . . . . . . . . . . . . 14 ⊢ seq1( + , 𝐹) ⇝ 1
154		climrel 15432	. . . . . . . . . . . . . . 15 ⊢ Rel ⇝
155	154	releldmi 5937	. . . . . . . . . . . . . 14 ⊢ (seq1( + , 𝐹) ⇝ 1 → seq1( + , 𝐹) ∈ dom ⇝ )
156	153, 155	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq1( + , 𝐹) ∈ dom ⇝ )
157	152, 156	eqeltrd 2825	. . . . . . . . . . . 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 12905	. . . . . . . . . . . . . . . 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 861	. . . . . . . . . . . . . 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 12903	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
166	164, 165	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 − 1) ∈ ℕ)
167		nncn 12216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
168	167	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 ∈ ℂ)
169		1cnd 11205	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 1 ∈ ℂ)
170	168, 169	npcand 11571	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
171	170	eqcomd 2730	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 = ((𝑁 − 1) + 1))
172	171	seqeq1d 13968	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) = seq((𝑁 − 1) + 1)( + , 𝐹))
173		nnuz 12861	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
174		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℕ)
175	138	recnd 11238	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
176	139, 175	eqeltrd 2825	. . . . . . . . . . . . . . . . 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 15597	. . . . . . . . . . . . . . 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 5160	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
182	154	releldmi 5937	. . . . . . . . . . . . 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 810	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → seq𝑁( + , 𝐹) ∈ dom ⇝ )
186	120, 56, 140, 151, 185	isumrecl 15707	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
187	121	nnrpd 13010	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ⁺)
188	187	rpge0d 13016	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ 𝑗)
189	141, 188	ge0p1rpd 13042	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ⁺)
190	187, 189	rpmulcld 13028	. . . . . . . . . . 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 13052	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ (1 / (𝑗 · (𝑗 + 1))))
193	120, 56, 140, 151, 185, 192	isumge0 15708	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))))
194	115, 186, 110, 193	leadd2dd 11825	. . . . . . . 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 11238	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
196	195	addridd 11410	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + 0) = Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))))
197	196	eqcomd 2730	. . . . . . . 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 11205	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 1 ∈ ℂ)
202	200, 201	addcld 11229	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℂ)
203	200, 202	mulcld 11230	. . . . . . . . . 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 11862	. . . . . . . . . 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 13	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → seq1( + , 𝐹) ∈ dom ⇝ )
209	173, 120, 198, 199, 207, 208	isumsplit 15782	. . . . . . . 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 5170	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))))
211		1zzd 12589	. . . . . . . . 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 15699	. . . . . . . 8 ⊢ (⊤ → Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1)
216	215	mptru 1540	. . . . . . 7 ⊢ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1
217	210, 216	breqtrdi 5179	. . . . . 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 12293	. . . . . . . 8 ⊢ 4 ∈ ℂ
220	219	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℂ)
221	112	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 4)
222	221	gt0ne0d 11774	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ≠ 0)
223	220, 222	reccld 11979	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℂ)
224	102	recnd 11238	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
225	61, 223, 224	fsummulc2 15726	. . . . 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 11227	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · 1) = (1 / 4))
227	218, 225, 226	3brtr3d 5169	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ≤ (1 / 4))
228	88, 104, 30, 109, 227	letrd 11367	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ (1 / 4))
229	60, 228	eqbrtrd 5160	. 2 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) ≤ (1 / 4))
230	17, 26, 30, 229	subled 11813	1 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2932 class class class wbr 5138 ↦ cmpt 5221 dom cdm 5666 ‘cfv 6533 (class class class)co 7401 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 4c4 12265 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ...cfz 13480 ..^cfzo 13623 seqcseq 13962 ↑cexp 14023 !cfa 14229 √csqrt 15176 ⇝ cli 15424 Σcsu 15628 eceu 16002 logclog 26404

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-e 16008 df-sin 16009 df-cos 16010 df-tan 16011 df-pi 16012 df-dvds 16194 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-cmp 23212 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 df-ulm 26229 df-log 26406 df-cxp 26407

This theorem is referenced by: stirlinglem13 45253

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