stirlinglem12 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem12 47231

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 12353	. . . . 5 ⊢ 1 ∈ ℕ
2		stirlinglem12.1	. . . . . . 7 ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
3	2	stirlinglem2 47221	. . . . . 6 ⊢ (1 ∈ ℕ → (𝐴‘1) ∈ ℝ⁺)
4		relogcl 26912	. . . . . 6 ⊢ ((𝐴‘1) ∈ ℝ⁺ → (log‘(𝐴‘1)) ∈ ℝ)
5	1, 3, 4	mp2b 10	. . . . 5 ⊢ (log‘(𝐴‘1)) ∈ ℝ
6		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑛1
7		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑛log
8		nfmpt1 5203	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))
9	2, 8	nfcxfr 2920	. . . . . . . 8 ⊢ Ⅎ𝑛𝐴
10	9, 6	nffv 6900	. . . . . . 7 ⊢ Ⅎ𝑛(𝐴‘1)
11	7, 10	nffv 6900	. . . . . 6 ⊢ Ⅎ𝑛(log‘(𝐴‘1))
12		2fveq3 6895	. . . . . 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 7020	. . . . 5 ⊢ ((1 ∈ ℕ ∧ (log‘(𝐴‘1)) ∈ ℝ) → (𝐵‘1) = (log‘(𝐴‘1)))
15	1, 5, 14	mp2an 705	. . . 4 ⊢ (𝐵‘1) = (log‘(𝐴‘1))
16	15, 5	eqeltri 2856	. . 3 ⊢ (𝐵‘1) ∈ ℝ
17	16	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘1) ∈ ℝ)
18	2	stirlinglem2 47221	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐴‘𝑁) ∈ ℝ⁺)
19	18	relogcld 26960	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(𝐴‘𝑁)) ∈ ℝ)
20		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑛𝑁
21	9, 20	nffv 6900	. . . . . 6 ⊢ Ⅎ𝑛(𝐴‘𝑁)
22	7, 21	nffv 6900	. . . . 5 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑁))
23		2fveq3 6895	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑁)))
24	20, 22, 23, 13	fvmptf 7020	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (log‘(𝐴‘𝑁)) ∈ ℝ) → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
25	19, 24	mpdan 700	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) = (log‘(𝐴‘𝑁)))
26	25, 19	eqeltrd 2860	. 2 ⊢ (𝑁 ∈ ℕ → (𝐵‘𝑁) ∈ ℝ)
27		4re 12434	. . . 4 ⊢ 4 ∈ ℝ
28		4ne0 12461	. . . 4 ⊢ 4 ≠ 0
29	27, 28	rereccli 12089	. . 3 ⊢ (1 / 4) ∈ ℝ
30	29	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℝ)
31		fveq2 6890	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
32		fveq2 6890	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝐵‘𝑘) = (𝐵‘(𝑗 + 1)))
33		fveq2 6890	. . . . 5 ⊢ (𝑘 = 1 → (𝐵‘𝑘) = (𝐵‘1))
34		fveq2 6890	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐵‘𝑘) = (𝐵‘𝑁))
35		elnnuz 13012	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
36	35	biimpi 219	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
37		elfznn 13694	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
38	2	stirlinglem2 47221	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ∈ ℝ⁺)
39	37, 38	syl 18	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℝ⁺)
40	39	relogcld 26960	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (log‘(𝐴‘𝑘)) ∈ ℝ)
41		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑘
42	9, 41	nffv 6900	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝐴‘𝑘)
43	7, 42	nffv 6900	. . . . . . . . 9 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑘))
44		2fveq3 6895	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑘)))
45	41, 43, 44, 13	fvmptf 7020	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (log‘(𝐴‘𝑘)) ∈ ℝ) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
46	37, 40, 45	syl2anc 596	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
47	46	adantl 487	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘)))
48	39	rpcnd 13173	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ∈ ℂ)
49	48	adantl 487	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
50	38	rpne0d 13176	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ≠ 0)
51	37, 50	syl 18	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴‘𝑘) ≠ 0)
52	51	adantl 487	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ≠ 0)
53	49, 52	logcld 26907	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘(𝐴‘𝑘)) ∈ ℂ)
54	47, 53	eqeltrd 2860	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℂ)
55	31, 32, 33, 34, 36, 54	telfsumo 15978	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = ((𝐵‘1) − (𝐵‘𝑁)))
56		nnz 12721	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
57		fzoval 13801	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (1..^𝑁) = (1...(𝑁 − 1)))
58	56, 57	syl 18	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1..^𝑁) = (1...(𝑁 − 1)))
59	58	sumeq1d 15876	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1..^𝑁)((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
60	55, 59	eqtr3d 2797	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) = Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))))
61		fzfid 14123	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
62		elfznn 13694	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℕ)
63	62	adantl 487	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → 𝑗 ∈ ℕ)
64	2	stirlinglem2 47221	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝐴‘𝑗) ∈ ℝ⁺)
65	64	relogcld 26960	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘𝑗)) ∈ ℝ)
66		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑗
67	9, 66	nffv 6900	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝐴‘𝑗)
68	7, 67	nffv 6900	. . . . . . . . . 10 ⊢ Ⅎ𝑛(log‘(𝐴‘𝑗))
69		2fveq3 6895	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑗)))
70	66, 68, 69, 13	fvmptf 7020	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ (log‘(𝐴‘𝑗)) ∈ ℝ) → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
71	65, 70	mpdan 700	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) = (log‘(𝐴‘𝑗)))
72	71, 65	eqeltrd 2860	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) ∈ ℝ)
73	63, 72	syl 18	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝐵‘𝑗) ∈ ℝ)
74		peano2nn 12354	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
75	2	stirlinglem2 47221	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
76	74, 75	syl 18	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈ ℝ⁺)
77	76	relogcld 26960	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ)
78		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑗 + 1)
79	9, 78	nffv 6900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝐴‘(𝑗 + 1))
80	7, 79	nffv 6900	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(log‘(𝐴‘(𝑗 + 1)))
81		2fveq3 6895	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘(𝑗 + 1))))
82	78, 80, 81, 13	fvmptf 7020	. . . . . . . . . 10 ⊢ (((𝑗 + 1) ∈ ℕ ∧ (log‘(𝐴‘(𝑗 + 1))) ∈ ℝ) → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1))))
83	74, 77, 82	syl2anc 596	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1))))
84	83, 77	eqeltrd 2860	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) ∈ ℝ)
85	62, 84	syl 18	. . . . . . 7 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
86	85	adantl 487	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
87	73, 86	resubcld 11751	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ∈ ℝ)
88	61, 87	fsumrecl 15909	. . . 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 12357	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℝ)
91		1red 11316	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℝ)
92	90, 91	readdcld 11345	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℝ)
93	90, 92	remulcld 11346	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
94	90	recnd 11344	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ∈ ℂ)
95		1cnd 11309	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 1 ∈ ℂ)
96	94, 95	addcld 11335	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 + 1) ∈ ℂ)
97	62	nnne0d 12395	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → 𝑗 ≠ 0)
98	74	nnne0d 12395	. . . . . . . . . 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 11975	. . . . . . . 8 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (𝑗 · (𝑗 + 1)) ≠ 0)
101	93, 100	rereccld 12151	. . . . . . 7 ⊢ (𝑗 ∈ (1...(𝑁 − 1)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
102	101	adantl 487	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
103	89, 102	remulcld 11346	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → ((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
104	61, 103	fsumrecl 15909	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ∈ ℝ)
105		eqid 2760	. . . . . . 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 2760	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖)) = (𝑖 ∈ ℕ ↦ ((1 / (((2 · 𝑗) + 1)↑2))↑𝑖))
107	2, 13, 105, 106	stirlinglem10 47229	. . . . . 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 15975	. . . 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 15909	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
111		1red 11316	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
112		4pos 12460	. . . . . . . . 9 ⊢ 0 < 4
113	27, 112	elrpii 13130	. . . . . . . 8 ⊢ 4 ∈ ℝ⁺
114	113	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℝ⁺)
115		0red 11318	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
116		0lt1 11845	. . . . . . . . 9 ⊢ 0 < 1
117	116	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 1)
118	115, 111, 117	ltled 11465	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ 1)
119	111, 114, 118	divge0d 13211	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / 4))
120		eqid 2760	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
121		eluznn 13052	. . . . . . . . . . 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 490	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → 𝑛 = 𝑗)
125	124	oveq1d 7439	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 + 1) = (𝑗 + 1))
126	124, 125	oveq12d 7442	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
127	126	oveq2d 7440	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = 𝑗) → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
128		id 23	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
129		nnre 12349	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
130		1red 11316	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℝ)
131	129, 130	readdcld 11345	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℝ)
132	129, 131	remulcld 11346	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ∈ ℝ)
133		nncn 12350	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
134		1cnd 11309	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 1 ∈ ℂ)
135	133, 134	addcld 11335	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℂ)
136		nnne0 12379	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → 𝑗 ≠ 0)
137	133, 135, 136, 98	mulne0d 11975	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 · (𝑗 + 1)) ≠ 0)
138	132, 137	rereccld 12151	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
139	123, 127, 128, 138	fvmptd 7006	. . . . . . . . . . 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 12357	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ)
142		1red 11316	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℝ)
143	141, 142	readdcld 11345	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ)
144	141, 143	remulcld 11346	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ∈ ℝ)
145	141	recnd 11344	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℂ)
146		1cnd 11309	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 1 ∈ ℂ)
147	145, 146	addcld 11335	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℂ)
148	121	nnne0d 12395	. . . . . . . . . . . 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 11975	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ≠ 0)
151	144, 150	rereccld 12151	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
152		seqeq1 14154	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq𝑁( + , 𝐹) = seq1( + , 𝐹))
153	122	trireciplem 16040	. . . . . . . . . . . . . 14 ⊢ seq1( + , 𝐹) ⇝ 1
154		climrel 15668	. . . . . . . . . . . . . . 15 ⊢ Rel ⇝
155	154	releldmi 5931	. . . . . . . . . . . . . 14 ⊢ (seq1( + , 𝐹) ⇝ 1 → seq1( + , 𝐹) ∈ dom ⇝ )
156	153, 155	mp1i 14	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → seq1( + , 𝐹) ∈ dom ⇝ )
157	152, 156	eqeltrd 2860	. . . . . . . . . . . 12 ⊢ (𝑁 = 1 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
158	157	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
159		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ ℕ)
160		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ¬ 𝑁 = 1)
161		elnn1uz2 13059	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
162	161	birani 509	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
163	162	ord 878	. . . . . . . . . . . . . 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 13057	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
166	164, 165	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (𝑁 − 1) ∈ ℕ)
167		nncn 12350	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
168	167	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 ∈ ℂ)
169		1cnd 11309	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 1 ∈ ℂ)
170	168, 169	npcand 11680	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
171	170	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → 𝑁 = ((𝑁 − 1) + 1))
172	171	seqeq1d 14157	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) = seq((𝑁 − 1) + 1)( + , 𝐹))
173		nnuz 13011	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
174		id 23	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℕ)
175	138	recnd 11344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
176	139, 175	eqeltrd 2860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) ∈ ℂ)
177	176	adantl 487	. . . . . . . . . . . . . . . 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 15831	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ ℕ → seq((𝑁 − 1) + 1)( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
180	179	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq((𝑁 − 1) + 1)( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
181	172, 180	eqbrtrd 5126	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 − 1) ∈ ℕ) → seq𝑁( + , 𝐹) ⇝ (1 − (seq1( + , 𝐹)‘(𝑁 − 1))))
182	154	releldmi 5931	. . . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → seq𝑁( + , 𝐹) ∈ dom ⇝ )
185	158, 184	pm2.61dan 825	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → seq𝑁( + , 𝐹) ∈ dom ⇝ )
186	120, 56, 140, 151, 185	isumrecl 15940	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
187	121	nnrpd 13169	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 𝑗 ∈ ℝ⁺)
188	187	rpge0d 13175	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ 𝑗)
189	141, 188	ge0p1rpd 13201	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 + 1) ∈ ℝ⁺)
190	187, 189	rpmulcld 13187	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (𝑗 · (𝑗 + 1)) ∈ ℝ⁺)
191	118	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ 1)
192	142, 190, 191	divge0d 13211	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → 0 ≤ (1 / (𝑗 · (𝑗 + 1))))
193	120, 56, 140, 151, 185, 192	isumge0 15941	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑗 ∈ (ℤ_≥‘𝑁)(1 / (𝑗 · (𝑗 + 1))))
194	115, 186, 110, 193	leadd2dd 11938	. . . . . . . 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 11344	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
196	195	addridd 11517	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) + 0) = Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))))
197	196	eqcomd 2766	. . . . . . . 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 487	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
200	133	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
201		1cnd 11309	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 1 ∈ ℂ)
202	200, 201	addcld 11335	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℂ)
203	200, 202	mulcld 11336	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
204	136	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ≠ 0)
205	98	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ≠ 0)
206	200, 202, 204, 205	mulne0d 11975	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 · (𝑗 + 1)) ≠ 0)
207	203, 206	reccld 12093	. . . . . . . . 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 16018	. . . . . . . 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 5136	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))))
211		1zzd 12734	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℤ)
212	139	adantl 487	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
213	175	adantl 487	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
214	153	a1i 11	. . . . . . . . 9 ⊢ (⊤ → seq1( + , 𝐹) ⇝ 1)
215	173, 211, 212, 213, 214	isumclim 15932	. . . . . . . 8 ⊢ (⊤ → Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1)
216	215	mptru 1576	. . . . . . 7 ⊢ Σ𝑗 ∈ ℕ (1 / (𝑗 · (𝑗 + 1))) = 1
217	210, 216	breqtrdi 5145	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1))) ≤ 1)
218	110, 111, 30, 119, 217	lemul2ad 12264	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · Σ𝑗 ∈ (1...(𝑁 − 1))(1 / (𝑗 · (𝑗 + 1)))) ≤ ((1 / 4) · 1))
219		4cn 12435	. . . . . . . 8 ⊢ 4 ∈ ℂ
220	219	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ∈ ℂ)
221	112	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 4)
222	221	gt0ne0d 11887	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 4 ≠ 0)
223	220, 222	reccld 12093	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / 4) ∈ ℂ)
224	102	recnd 11344	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
225	61, 223, 224	fsummulc2 15959	. . . . 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 11333	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 / 4) · 1) = (1 / 4))
227	218, 225, 226	3brtr3d 5135	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((1 / 4) · (1 / (𝑗 · (𝑗 + 1)))) ≤ (1 / 4))
228	88, 104, 30, 109, 227	letrd 11474	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑗 ∈ (1...(𝑁 − 1))((𝐵‘𝑗) − (𝐵‘(𝑗 + 1))) ≤ (1 / 4))
229	60, 228	eqbrtrd 5126	. 2 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (𝐵‘𝑁)) ≤ (1 / 4))
230	17, 26, 30, 229	subled 11926	1 ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ⊤wtru 1570 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5651 ‘cfv 6543 (class class class)co 7424 ℂcc 11205 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 · cmul 11212 < clt 11350 ≤ cle 11351 − cmin 11548 / cdiv 11980 ℕcn 12342 2c2 12404 4c4 12406 ℤcz 12700 ℤ_≥cuz 12972 ℝ⁺crp 13127 ...cfz 13646 ..^cfzo 13795 seqcseq 14151 ↑cexp 14211 !cfa 14424 √csqrt 15409 ⇝ cli 15660 Σcsu 15862 eceu 16237 logclog 26891

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 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 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-oadd 8488 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-xnn0 12687 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-fac 14425 df-bc 14454 df-hash 14482 df-shft 15229 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-limsup 15647 df-clim 15664 df-rlim 15665 df-sum 15863 df-ef 16242 df-e 16243 df-sin 16244 df-cos 16245 df-tan 16246 df-pi 16247 df-dvds 16432 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-cmp 23714 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196 df-ulm 26713 df-log 26893 df-cxp 26894

This theorem is used by: stirlinglem13 47232

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