logfaclbnd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > logfaclbnd		Structured version Visualization version GIF version

Theorem logfaclbnd 27185

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 12953	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
2	1	times2d 12421	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 2) = (𝐴 + 𝐴))
3	2	oveq2d 7383	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − (𝐴 · 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 + 𝐴)))
4		relogcl 26539	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 11173	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6		2cnd 12259	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
7	1, 5, 6	subdid 11606	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 · 2)))
8		rpre 12951	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
9	8, 4	remulcld 11175	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℝ)
10	9	recnd 11173	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℂ)
11	10, 1, 1	subsub4d 11536	. . 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 11578	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ∈ ℝ)
14		fzfid 13935	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
15		fzfid 13935	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
16		elfznn 13507	. . . . . . . 8 ⊢ (𝑑 ∈ (1...𝑛) → 𝑑 ∈ ℕ)
17	16	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → 𝑑 ∈ ℕ)
18	17	nnrecred 12228	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℝ)
19	15, 18	fsumrecl 15696	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
20	14, 19	fsumrecl 15696	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
21		rprege0 12958	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
22		flge0nn0 13779	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
24	23	faccld 14246	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℕ)
25	24	nnrpd 12984	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
26	25	relogcld 26587	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
27	26, 8	readdcld 11174	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + 𝐴) ∈ ℝ)
28		elfznn 13507	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℕ)
29	28	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
30	29	nnrecred 12228	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
31	14, 30	fsumrecl 15696	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
32	8, 31	remulcld 11175	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) ∈ ℝ)
33		reflcl 13755	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
34	8, 33	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
35	32, 34	resubcld 11578	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) ∈ ℝ)
36		harmoniclbnd 26972	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))
37		rpregt0 12957	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
38		lemul2 12008	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ↔ (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))))
40	36, 39	mpbid 232	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)))
41		flle 13758	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
42	8, 41	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
43	9, 34, 32, 8, 40, 42	le2subd 11770	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)))
44	28	nnrecred 12228	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (1 / 𝑑) ∈ ℝ)
45		remulcl 11123	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 𝑑) ∈ ℝ) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
46	8, 44, 45	syl2an 597	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
47		peano2rem 11461	. . . . . . . 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 13935	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑑...(⌊‘𝐴)) ∈ Fin)
50	30	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
51	49, 50	fsumrecl 15696	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
52	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
53	52, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℝ)
54		peano2re 11319	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℝ → ((⌊‘𝐴) + 1) ∈ ℝ)
55	53, 54	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((⌊‘𝐴) + 1) ∈ ℝ)
56	29	nnred 12189	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ)
57		fllep1 13760	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
58	8, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
59	58	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ≤ ((⌊‘𝐴) + 1))
60	52, 55, 56, 59	lesub1dd 11766	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑))
61	52, 56	resubcld 11578	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ∈ ℝ)
62	55, 56	resubcld 11578	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) ∈ ℝ)
63	29	nnrpd 12984	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ⁺)
64	63	rpreccld 12996	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ⁺)
65	61, 62, 64	lemul1d 13029	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑) ↔ ((𝐴 − 𝑑) · (1 / 𝑑)) ≤ ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑))))
66	60, 65	mpbid 232	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) ≤ ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
67	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℂ)
68	29	nncnd 12190	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℂ)
69	30	recnd 11173	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℂ)
70	67, 68, 69	subdird 11607	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) = ((𝐴 · (1 / 𝑑)) − (𝑑 · (1 / 𝑑))))
71	29	nnne0d 12227	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ≠ 0)
72	68, 71	recidd 11926	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑑 · (1 / 𝑑)) = 1)
73	72	oveq2d 7383	. . . . . . . . 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 15752	. . . . . . . . . 10 ⊢ (((𝑑...(⌊‘𝐴)) ∈ Fin ∧ (1 / 𝑑) ∈ ℂ) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
76	49, 69, 75	syl2anc 585	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
77		elfzuz3 13475	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
78	77	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
79		hashfz 14389	. . . . . . . . . . . 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 11173	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
82	81	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℂ)
83		1cnd 11139	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
84	82, 83, 68	addsubd 11526	. . . . . . . . . . 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 7382	. . . . . . . . 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 5118	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
89	14, 48, 51, 88	fsumle 15762	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
90	14, 1, 69	fsummulc2 15746	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)))
91		ax-1cn 11096	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
92		fsumconst 15752	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
93	14, 91, 92	sylancl 587	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
94		hashfz1 14308	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
95	23, 94	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
96	95	oveq1d 7382	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((♯‘(1...(⌊‘𝐴))) · 1) = ((⌊‘𝐴) · 1))
97	81	mulridd 11162	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) · 1) = (⌊‘𝐴))
98	93, 96, 97	3eqtrrd 2777	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))1)
99	90, 98	oveq12d 7385	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)) − Σ𝑑 ∈ (1...(⌊‘𝐴))1))
100	46	recnd 11173	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℂ)
101	14, 100, 83	fsumsub 15750	. . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) = (ℤ_≥‘1)
104	103	uztrn2 12807	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)) → 𝑛 ∈ (ℤ_≥‘1))
105	104	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → 𝑛 ∈ (ℤ_≥‘1))
106	105	biantrurd 532	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
107		uzss 12811	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑑) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
108	107	ad2antll 730	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
109	108	sseld 3921	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
110	109	pm4.71rd 562	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) ↔ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
111	106, 110	bitr3d 281	. . . . . . . . . 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 460	. . . . . . . . 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 657	. . . . . . . . 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 314	. . . . . . . 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 13472	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
117		elfzuzb 13472	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...𝑛) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)))
118	116, 117	anbi12i 629	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ ((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))))
119		elfzuzb 13472	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
120		elfzuzb 13472	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑑...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
121	119, 120	anbi12i 629	. . . . . . . 8 ⊢ ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴))) ↔ ((𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)) ∧ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
122	115, 118, 121	3bitr4g 314	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴)))))
123	18	recnd 11173	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℂ)
124	123	anasss 466	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛))) → (1 / 𝑑) ∈ ℂ)
125	14, 14, 15, 122, 124	fsumcom2 15736	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
126	89, 102, 125	3brtr4d 5118	. . . . 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 11303	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
128	26, 34	readdcld 11174	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ∈ ℝ)
129		elfznn 13507	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
130	129	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
131	130	nnrpd 12984	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
132	131	relogcld 26587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
133		peano2re 11319	. . . . . . . 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 12545	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
136		flid 13767	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (⌊‘𝑛) = 𝑛)
137	135, 136	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (⌊‘𝑛) = 𝑛)
138	137	oveq2d 7383	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1...(⌊‘𝑛)) = (1...𝑛))
139	138	sumeq1d 15662	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) = Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
140		nnre 12181	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
141		nnge1 12205	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ≤ 𝑛)
142		harmonicubnd 26973	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ 1 ≤ 𝑛) → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
143	140, 141, 142	syl2anc 585	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
144	139, 143	eqbrtrrd 5110	. . . . . . . 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 15762	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1))
147	132	recnd 11173	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℂ)
148		1cnd 11139	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
149	14, 147, 148	fsumadd 15702	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) + Σ𝑛 ∈ (1...(⌊‘𝐴))1))
150		logfac 26565	. . . . . . . . 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 15752	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
153	14, 91, 152	sylancl 587	. . . . . . . . 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 7385	. . . . . . 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 5112	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
158	34, 8, 26, 42	leadd2dd 11765	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
159	20, 128, 27, 157, 158	letrd 11303	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
160	13, 20, 27, 127, 159	letrd 11303	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
161	13, 8, 26	lesubaddd 11747	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) ≤ (log‘(!‘(⌊‘𝐴))) ↔ ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴)))
162	160, 161	mpbird 257	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) ≤ (log‘(!‘(⌊‘𝐴))))
163	12, 162	eqbrtrd 5108	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 class class class wbr 5086 ‘cfv 6499 (class class class)co 7367 Fincfn 8893 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ℝ⁺crp 12942 ...cfz 13461 ⌊cfl 13749 !cfa 14235 ♯chash 14292 Σcsu 15648 logclog 26518

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-e 16033 df-sin 16034 df-cos 16035 df-tan 16036 df-pi 16037 df-dvds 16222 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-ulm 26342 df-log 26520 df-atan 26831 df-em 26956

This theorem is referenced by: (None)

W3C validator