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Theorem logfaclbnd 26714

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 12980	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
2	1	times2d 12452	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 2) = (𝐴 + 𝐴))
3	2	oveq2d 7421	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − (𝐴 · 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 + 𝐴)))
4		relogcl 26075	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 11238	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6		2cnd 12286	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
7	1, 5, 6	subdid 11666	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 · 2)))
8		rpre 12978	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
9	8, 4	remulcld 11240	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℝ)
10	9	recnd 11238	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℂ)
11	10, 1, 1	subsub4d 11598	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) = ((𝐴 · (log‘𝐴)) − (𝐴 + 𝐴)))
12	3, 7, 11	3eqtr4d 2782	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴))
13	9, 8	resubcld 11638	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ∈ ℝ)
14		fzfid 13934	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
15		fzfid 13934	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
16		elfznn 13526	. . . . . . . 8 ⊢ (𝑑 ∈ (1...𝑛) → 𝑑 ∈ ℕ)
17	16	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → 𝑑 ∈ ℕ)
18	17	nnrecred 12259	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℝ)
19	15, 18	fsumrecl 15676	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
20	14, 19	fsumrecl 15676	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
21		rprege0 12985	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
22		flge0nn0 13781	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
24	23	faccld 14240	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℕ)
25	24	nnrpd 13010	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
26	25	relogcld 26122	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
27	26, 8	readdcld 11239	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + 𝐴) ∈ ℝ)
28		elfznn 13526	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℕ)
29	28	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
30	29	nnrecred 12259	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
31	14, 30	fsumrecl 15676	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
32	8, 31	remulcld 11240	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) ∈ ℝ)
33		reflcl 13757	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
34	8, 33	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
35	32, 34	resubcld 11638	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) ∈ ℝ)
36		harmoniclbnd 26502	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))
37		rpregt0 12984	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
38		lemul2 12063	. . . . . . . 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 1371	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ↔ (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))))
40	36, 39	mpbid 231	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)))
41		flle 13760	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
42	8, 41	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
43	9, 34, 32, 8, 40, 42	le2subd 11830	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)))
44	28	nnrecred 12259	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (1 / 𝑑) ∈ ℝ)
45		remulcl 11191	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 𝑑) ∈ ℝ) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
46	8, 44, 45	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
47		peano2rem 11523	. . . . . . . 8 ⊢ ((𝐴 · (1 / 𝑑)) ∈ ℝ → ((𝐴 · (1 / 𝑑)) − 1) ∈ ℝ)
48	46, 47	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) ∈ ℝ)
49		fzfid 13934	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑑...(⌊‘𝐴)) ∈ Fin)
50	30	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
51	49, 50	fsumrecl 15676	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
52	8	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
53	52, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℝ)
54		peano2re 11383	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℝ → ((⌊‘𝐴) + 1) ∈ ℝ)
55	53, 54	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((⌊‘𝐴) + 1) ∈ ℝ)
56	29	nnred 12223	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ)
57		fllep1 13762	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
58	8, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
59	58	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ≤ ((⌊‘𝐴) + 1))
60	52, 55, 56, 59	lesub1dd 11826	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑))
61	52, 56	resubcld 11638	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ∈ ℝ)
62	55, 56	resubcld 11638	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) ∈ ℝ)
63	29	nnrpd 13010	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ⁺)
64	63	rpreccld 13022	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ⁺)
65	61, 62, 64	lemul1d 13055	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑) ↔ ((𝐴 − 𝑑) · (1 / 𝑑)) ≤ ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑))))
66	60, 65	mpbid 231	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) ≤ ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
67	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℂ)
68	29	nncnd 12224	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℂ)
69	30	recnd 11238	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℂ)
70	67, 68, 69	subdird 11667	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) = ((𝐴 · (1 / 𝑑)) − (𝑑 · (1 / 𝑑))))
71	29	nnne0d 12258	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ≠ 0)
72	68, 71	recidd 11981	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑑 · (1 / 𝑑)) = 1)
73	72	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − (𝑑 · (1 / 𝑑))) = ((𝐴 · (1 / 𝑑)) − 1))
74	70, 73	eqtr2d 2773	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) = ((𝐴 − 𝑑) · (1 / 𝑑)))
75		fsumconst 15732	. . . . . . . . . 10 ⊢ (((𝑑...(⌊‘𝐴)) ∈ Fin ∧ (1 / 𝑑) ∈ ℂ) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
76	49, 69, 75	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
77		elfzuz3 13494	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
78	77	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
79		hashfz 14383	. . . . . . . . . . . 12 ⊢ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) − 𝑑) + 1))
80	78, 79	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) − 𝑑) + 1))
81	34	recnd 11238	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
82	81	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℂ)
83		1cnd 11205	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
84	82, 83, 68	addsubd 11588	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) = (((⌊‘𝐴) − 𝑑) + 1))
85	80, 84	eqtr4d 2775	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) + 1) − 𝑑))
86	85	oveq1d 7420	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)) = ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
87	76, 86	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
88	66, 74, 87	3brtr4d 5179	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
89	14, 48, 51, 88	fsumle 15741	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
90	14, 1, 69	fsummulc2 15726	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)))
91		ax-1cn 11164	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
92		fsumconst 15732	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
93	14, 91, 92	sylancl 586	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
94		hashfz1 14302	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
95	23, 94	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
96	95	oveq1d 7420	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((♯‘(1...(⌊‘𝐴))) · 1) = ((⌊‘𝐴) · 1))
97	81	mulridd 11227	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) · 1) = (⌊‘𝐴))
98	93, 96, 97	3eqtrrd 2777	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))1)
99	90, 98	oveq12d 7423	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)) − Σ𝑑 ∈ (1...(⌊‘𝐴))1))
100	46	recnd 11238	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℂ)
101	14, 100, 83	fsumsub 15730	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)) − Σ𝑑 ∈ (1...(⌊‘𝐴))1))
102	99, 101	eqtr4d 2775	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) = Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1))
103		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) = (ℤ_≥‘1)
104	103	uztrn2 12837	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)) → 𝑛 ∈ (ℤ_≥‘1))
105	104	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → 𝑛 ∈ (ℤ_≥‘1))
106	105	biantrurd 533	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
107		uzss 12841	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑑) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
108	107	ad2antll 727	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
109	108	sseld 3980	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
110	109	pm4.71rd 563	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) ↔ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
111	106, 110	bitr3d 280	. . . . . . . . . 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 579	. . . . . . . . 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 461	. . . . . . . . 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 654	. . . . . . . . 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 313	. . . . . . . 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 13491	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
117		elfzuzb 13491	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...𝑛) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)))
118	116, 117	anbi12i 627	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ ((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))))
119		elfzuzb 13491	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
120		elfzuzb 13491	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑑...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
121	119, 120	anbi12i 627	. . . . . . . 8 ⊢ ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴))) ↔ ((𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)) ∧ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
122	115, 118, 121	3bitr4g 313	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴)))))
123	18	recnd 11238	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℂ)
124	123	anasss 467	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛))) → (1 / 𝑑) ∈ ℂ)
125	14, 14, 15, 122, 124	fsumcom2 15716	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
126	89, 102, 125	3brtr4d 5179	. . . . 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 11367	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
128	26, 34	readdcld 11239	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ∈ ℝ)
129		elfznn 13526	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
130	129	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
131	130	nnrpd 13010	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
132	131	relogcld 26122	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
133		peano2re 11383	. . . . . . . 8 ⊢ ((log‘𝑛) ∈ ℝ → ((log‘𝑛) + 1) ∈ ℝ)
134	132, 133	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛) + 1) ∈ ℝ)
135		nnz 12575	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
136		flid 13769	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (⌊‘𝑛) = 𝑛)
137	135, 136	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (⌊‘𝑛) = 𝑛)
138	137	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1...(⌊‘𝑛)) = (1...𝑛))
139	138	sumeq1d 15643	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) = Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
140		nnre 12215	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
141		nnge1 12236	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ≤ 𝑛)
142		harmonicubnd 26503	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ 1 ≤ 𝑛) → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
143	140, 141, 142	syl2anc 584	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
144	139, 143	eqbrtrrd 5171	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘𝑛) + 1))
145	130, 144	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘𝑛) + 1))
146	14, 19, 134, 145	fsumle 15741	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1))
147	132	recnd 11238	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℂ)
148		1cnd 11205	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
149	14, 147, 148	fsumadd 15682	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) + Σ𝑛 ∈ (1...(⌊‘𝐴))1))
150		logfac 26100	. . . . . . . . 9 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
151	23, 150	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
152		fsumconst 15732	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
153	14, 91, 152	sylancl 586	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
154	153, 96, 97	3eqtrrd 2777	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))1)
155	151, 154	oveq12d 7423	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) + Σ𝑛 ∈ (1...(⌊‘𝐴))1))
156	149, 155	eqtr4d 2775	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
157	146, 156	breqtrd 5173	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
158	34, 8, 26, 42	leadd2dd 11825	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
159	20, 128, 27, 157, 158	letrd 11367	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
160	13, 20, 27, 127, 159	letrd 11367	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
161	13, 8, 26	lesubaddd 11807	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) ≤ (log‘(!‘(⌊‘𝐴))) ↔ ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴)))
162	160, 161	mpbird 256	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) ≤ (log‘(!‘(⌊‘𝐴))))
163	12, 162	eqbrtrd 5169	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ...cfz 13480 ⌊cfl 13751 !cfa 14229 ♯chash 14286 Σcsu 15628 logclog 26054

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 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 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-ulm 25880 df-log 26056 df-atan 26361 df-em 26486

This theorem is referenced by: (None)

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