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Theorem logfaclbnd 27549

Description: A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

**Assertion**
Ref	Expression
logfaclbnd	⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴))))

Proof of Theorem logfaclbnd Dummy variables 𝑑 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpcn 13131	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
2	1	times2d 12590	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 2) = (𝐴 + 𝐴))
3	2	oveq2d 7436	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − (𝐴 · 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 + 𝐴)))
4		relogcl 26903	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 11337	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6		2cnd 12421	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
7	1, 5, 6	subdid 11772	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 · 2)))
8		rpre 13129	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
9	8, 4	remulcld 11339	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℝ)
10	9	recnd 11337	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℂ)
11	10, 1, 1	subsub4d 11700	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) = ((𝐴 · (log‘𝐴)) − (𝐴 + 𝐴)))
12	3, 7, 11	3eqtr4d 2805	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴))
13	9, 8	resubcld 11744	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ∈ ℝ)
14		fzfid 14116	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
15		fzfid 14116	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
16		elfznn 13687	. . . . . . . 8 ⊢ (𝑑 ∈ (1...𝑛) → 𝑑 ∈ ℕ)
17	16	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → 𝑑 ∈ ℕ)
18	17	nnrecred 12389	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℝ)
19	15, 18	fsumrecl 15900	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
20	14, 19	fsumrecl 15900	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
21		rprege0 13136	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
22		flge0nn0 13960	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
23	21, 22	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
24	23	faccld 14428	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℕ)
25	24	nnrpd 13162	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
26	25	relogcld 26951	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
27	26, 8	readdcld 11338	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + 𝐴) ∈ ℝ)
28		elfznn 13687	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℕ)
29	28	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
30	29	nnrecred 12389	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
31	14, 30	fsumrecl 15900	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
32	8, 31	remulcld 11339	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) ∈ ℝ)
33		reflcl 13936	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
34	8, 33	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
35	32, 34	resubcld 11744	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) ∈ ℝ)
36		harmoniclbnd 27336	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))
37		rpregt0 13135	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
38		lemul2 12170	. . . . . . . 8 ⊢ (((log‘𝐴) ∈ ℝ ∧ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → ((log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ↔ (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))))
39	4, 31, 37, 38	syl3anc 1398	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ↔ (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))))
40	36, 39	mpbid 235	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)))
41		flle 13939	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
42	8, 41	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
43	9, 34, 32, 8, 40, 42	le2subd 11936	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)))
44	28	nnrecred 12389	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (1 / 𝑑) ∈ ℝ)
45		remulcl 11285	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 𝑑) ∈ ℝ) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
46	8, 44, 45	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
47		peano2rem 11625	. . . . . . . 8 ⊢ ((𝐴 · (1 / 𝑑)) ∈ ℝ → ((𝐴 · (1 / 𝑑)) − 1) ∈ ℝ)
48	46, 47	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) ∈ ℝ)
49		fzfid 14116	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑑...(⌊‘𝐴)) ∈ Fin)
50	30	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
51	49, 50	fsumrecl 15900	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
52	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
53	52, 33	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℝ)
54		peano2re 11483	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℝ → ((⌊‘𝐴) + 1) ∈ ℝ)
55	53, 54	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((⌊‘𝐴) + 1) ∈ ℝ)
56	29	nnred 12350	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ)
57		fllep1 13941	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
58	8, 57	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
59	58	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ≤ ((⌊‘𝐴) + 1))
60	52, 55, 56, 59	lesub1dd 11932	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑))
61	52, 56	resubcld 11744	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ∈ ℝ)
62	55, 56	resubcld 11744	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) ∈ ℝ)
63	29	nnrpd 13162	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ⁺)
64	63	rpreccld 13174	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ⁺)
65	61, 62, 64	lemul1d 13207	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑) ↔ ((𝐴 − 𝑑) · (1 / 𝑑)) ≤ ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑))))
66	60, 65	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) ≤ ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
67	1	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℂ)
68	29	nncnd 12351	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℂ)
69	30	recnd 11337	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℂ)
70	67, 68, 69	subdird 11773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) = ((𝐴 · (1 / 𝑑)) − (𝑑 · (1 / 𝑑))))
71	29	nnne0d 12388	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ≠ 0)
72	68, 71	recidd 12088	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑑 · (1 / 𝑑)) = 1)
73	72	oveq2d 7436	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − (𝑑 · (1 / 𝑑))) = ((𝐴 · (1 / 𝑑)) − 1))
74	70, 73	eqtr2d 2796	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) = ((𝐴 − 𝑑) · (1 / 𝑑)))
75		fsumconst 15956	. . . . . . . . . 10 ⊢ (((𝑑...(⌊‘𝐴)) ∈ Fin ∧ (1 / 𝑑) ∈ ℂ) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
76	49, 69, 75	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
77		elfzuz3 13653	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
78	77	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
79		hashfz 14572	. . . . . . . . . . . 12 ⊢ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) − 𝑑) + 1))
80	78, 79	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) − 𝑑) + 1))
81	34	recnd 11337	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
82	81	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℂ)
83		1cnd 11302	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
84	82, 83, 68	addsubd 11690	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) = (((⌊‘𝐴) − 𝑑) + 1))
85	80, 84	eqtr4d 2798	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) + 1) − 𝑑))
86	85	oveq1d 7435	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)) = ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
87	76, 86	eqtrd 2795	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
88	66, 74, 87	3brtr4d 5136	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
89	14, 48, 51, 88	fsumle 15966	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
90	14, 1, 69	fsummulc2 15950	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)))
91		ax-1cn 11258	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
92		fsumconst 15956	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
93	14, 91, 92	sylancl 598	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
94		hashfz1 14490	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
95	23, 94	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
96	95	oveq1d 7435	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((♯‘(1...(⌊‘𝐴))) · 1) = ((⌊‘𝐴) · 1))
97	81	mulridd 11326	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) · 1) = (⌊‘𝐴))
98	93, 96, 97	3eqtrrd 2800	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))1)
99	90, 98	oveq12d 7438	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)) − Σ𝑑 ∈ (1...(⌊‘𝐴))1))
100	46	recnd 11337	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℂ)
101	14, 100, 83	fsumsub 15954	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)) − Σ𝑑 ∈ (1...(⌊‘𝐴))1))
102	99, 101	eqtr4d 2798	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) = Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1))
103		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) = (ℤ_≥‘1)
104	103	uztrn2 12984	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)) → 𝑛 ∈ (ℤ_≥‘1))
105	104	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → 𝑛 ∈ (ℤ_≥‘1))
106	105	biantrurd 542	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
107		uzss 12988	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑑) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
108	107	ad2antll 742	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
109	108	sseld 3907	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
110	109	pm4.71rd 572	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) ↔ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
111	106, 110	bitr3d 284	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ↔ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
112	111	pm5.32da 590	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (((𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)) ∧ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))) ↔ ((𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)) ∧ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))))
113		ancom 466	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) ↔ ((𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)) ∧ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
114		an4 669	. . . . . . . . 9 ⊢ (((𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)) ∧ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))) ↔ ((𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)) ∧ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
115	112, 113, 114	3bitr4g 317	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) ↔ ((𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)) ∧ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))))
116		elfzuzb 13650	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
117		elfzuzb 13650	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...𝑛) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)))
118	116, 117	anbi12i 640	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ ((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))))
119		elfzuzb 13650	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
120		elfzuzb 13650	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑑...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
121	119, 120	anbi12i 640	. . . . . . . 8 ⊢ ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴))) ↔ ((𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)) ∧ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
122	115, 118, 121	3bitr4g 317	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴)))))
123	18	recnd 11337	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℂ)
124	123	anasss 472	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛))) → (1 / 𝑑) ∈ ℂ)
125	14, 14, 15, 122, 124	fsumcom2 15940	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
126	89, 102, 125	3brtr4d 5136	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
127	13, 35, 20, 43, 126	letrd 11467	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
128	26, 34	readdcld 11338	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ∈ ℝ)
129		elfznn 13687	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
130	129	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
131	130	nnrpd 13162	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
132	131	relogcld 26951	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
133		peano2re 11483	. . . . . . . 8 ⊢ ((log‘𝑛) ∈ ℝ → ((log‘𝑛) + 1) ∈ ℝ)
134	132, 133	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛) + 1) ∈ ℝ)
135		nnz 12714	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
136		flid 13948	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (⌊‘𝑛) = 𝑛)
137	135, 136	syl 18	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (⌊‘𝑛) = 𝑛)
138	137	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1...(⌊‘𝑛)) = (1...𝑛))
139	138	sumeq1d 15867	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) = Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
140		nnre 12342	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
141		nnge1 12366	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ≤ 𝑛)
142		harmonicubnd 27337	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ 1 ≤ 𝑛) → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
143	140, 141, 142	syl2anc 596	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
144	139, 143	eqbrtrrd 5128	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘𝑛) + 1))
145	130, 144	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘𝑛) + 1))
146	14, 19, 134, 145	fsumle 15966	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1))
147	132	recnd 11337	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℂ)
148		1cnd 11302	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
149	14, 147, 148	fsumadd 15906	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) + Σ𝑛 ∈ (1...(⌊‘𝐴))1))
150		logfac 26929	. . . . . . . . 9 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
151	23, 150	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
152		fsumconst 15956	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
153	14, 91, 152	sylancl 598	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
154	153, 96, 97	3eqtrrd 2800	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))1)
155	151, 154	oveq12d 7438	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) + Σ𝑛 ∈ (1...(⌊‘𝐴))1))
156	149, 155	eqtr4d 2798	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
157	146, 156	breqtrd 5130	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
158	34, 8, 26, 42	leadd2dd 11931	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
159	20, 128, 27, 157, 158	letrd 11467	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
160	13, 20, 27, 127, 159	letrd 11467	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
161	13, 8, 26	lesubaddd 11913	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) ≤ (log‘(!‘(⌊‘𝐴))) ↔ ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴)))
162	160, 161	mpbird 260	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) ≤ (log‘(!‘(⌊‘𝐴))))
163	12, 162	eqbrtrd 5126	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 Fincfn 8973 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 < clt 11343 ≤ cle 11344 − cmin 11541 / cdiv 11973 ℕcn 12335 2c2 12397 ℕ₀cn0 12606 ℤcz 12693 ℤ_≥cuz 12965 ℝ⁺crp 13120 ...cfz 13639 ⌊cfl 13930 !cfa 14417 ♯chash 14474 Σcsu 15853 logclog 26882

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 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

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 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 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 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-e 16234 df-sin 16235 df-cos 16236 df-tan 16237 df-pi 16238 df-dvds 16423 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-ulm 26704 df-log 26884 df-atan 27195 df-em 27320

This theorem is used by: (None)

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