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Theorem logfaclbnd 25399

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 12149	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
2	1	times2d 11626	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 2) = (𝐴 + 𝐴))
3	2	oveq2d 6938	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − (𝐴 · 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 + 𝐴)))
4		relogcl 24759	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 10405	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6		2cnd 11453	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
7	1, 5, 6	subdid 10831	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 · 2)))
8		rpre 12145	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
9	8, 4	remulcld 10407	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℝ)
10	9	recnd 10405	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℂ)
11	10, 1, 1	subsub4d 10765	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) = ((𝐴 · (log‘𝐴)) − (𝐴 + 𝐴)))
12	3, 7, 11	3eqtr4d 2823	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴))
13	9, 8	resubcld 10803	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ∈ ℝ)
14		fzfid 13091	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
15		fzfid 13091	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
16		elfznn 12687	. . . . . . . 8 ⊢ (𝑑 ∈ (1...𝑛) → 𝑑 ∈ ℕ)
17	16	adantl 475	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → 𝑑 ∈ ℕ)
18	17	nnrecred 11426	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℝ)
19	15, 18	fsumrecl 14872	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
20	14, 19	fsumrecl 14872	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
21		rprege0 12154	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
22		flge0nn0 12940	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
24		faccl 13388	. . . . . . . 8 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (!‘(⌊‘𝐴)) ∈ ℕ)
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℕ)
26	25	nnrpd 12179	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
27	26	relogcld 24806	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
28	27, 8	readdcld 10406	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + 𝐴) ∈ ℝ)
29		elfznn 12687	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℕ)
30	29	adantl 475	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
31	30	nnrecred 11426	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
32	14, 31	fsumrecl 14872	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
33	8, 32	remulcld 10407	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) ∈ ℝ)
34		reflcl 12916	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
35	8, 34	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
36	33, 35	resubcld 10803	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) ∈ ℝ)
37		harmoniclbnd 25187	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))
38		rpregt0 12153	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
39		lemul2 11230	. . . . . . . 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))))
40	4, 32, 38, 39	syl3anc 1439	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ↔ (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))))
41	37, 40	mpbid 224	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ≤ (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)))
42		flle 12919	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
43	8, 42	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
44	9, 35, 33, 8, 41, 43	le2subd 10995	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)))
45	29	nnrecred 11426	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (1 / 𝑑) ∈ ℝ)
46		remulcl 10357	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 𝑑) ∈ ℝ) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
47	8, 45, 46	syl2an 589	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
48		peano2rem 10690	. . . . . . . 8 ⊢ ((𝐴 · (1 / 𝑑)) ∈ ℝ → ((𝐴 · (1 / 𝑑)) − 1) ∈ ℝ)
49	47, 48	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) ∈ ℝ)
50		fzfid 13091	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑑...(⌊‘𝐴)) ∈ Fin)
51	31	adantr 474	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
52	50, 51	fsumrecl 14872	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
53	8	adantr 474	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
54	53, 34	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℝ)
55		peano2re 10549	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℝ → ((⌊‘𝐴) + 1) ∈ ℝ)
56	54, 55	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((⌊‘𝐴) + 1) ∈ ℝ)
57	30	nnred 11391	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ)
58		fllep1 12921	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
59	8, 58	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
60	59	adantr 474	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ≤ ((⌊‘𝐴) + 1))
61	53, 56, 57, 60	lesub1dd 10991	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑))
62	53, 57	resubcld 10803	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ∈ ℝ)
63	56, 57	resubcld 10803	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) ∈ ℝ)
64	30	nnrpd 12179	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ⁺)
65	64	rpreccld 12191	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ⁺)
66	62, 63, 65	lemul1d 12224	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑) ↔ ((𝐴 − 𝑑) · (1 / 𝑑)) ≤ ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑))))
67	61, 66	mpbid 224	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) ≤ ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
68	1	adantr 474	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℂ)
69	30	nncnd 11392	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℂ)
70	31	recnd 10405	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℂ)
71	68, 69, 70	subdird 10832	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) = ((𝐴 · (1 / 𝑑)) − (𝑑 · (1 / 𝑑))))
72	30	nnne0d 11425	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ≠ 0)
73	69, 72	recidd 11146	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑑 · (1 / 𝑑)) = 1)
74	73	oveq2d 6938	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − (𝑑 · (1 / 𝑑))) = ((𝐴 · (1 / 𝑑)) − 1))
75	71, 74	eqtr2d 2814	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) = ((𝐴 − 𝑑) · (1 / 𝑑)))
76		fsumconst 14926	. . . . . . . . . 10 ⊢ (((𝑑...(⌊‘𝐴)) ∈ Fin ∧ (1 / 𝑑) ∈ ℂ) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
77	50, 70, 76	syl2anc 579	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
78		elfzuz3 12656	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
79	78	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
80		hashfz 13528	. . . . . . . . . . . 12 ⊢ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) − 𝑑) + 1))
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) − 𝑑) + 1))
82	35	recnd 10405	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
83	82	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℂ)
84		1cnd 10371	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
85	83, 84, 69	addsubd 10755	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) = (((⌊‘𝐴) − 𝑑) + 1))
86	81, 85	eqtr4d 2816	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (♯‘(𝑑...(⌊‘𝐴))) = (((⌊‘𝐴) + 1) − 𝑑))
87	86	oveq1d 6937	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)) = ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
88	77, 87	eqtrd 2813	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
89	67, 75, 88	3brtr4d 4918	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
90	14, 49, 52, 89	fsumle 14935	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
91	14, 1, 70	fsummulc2 14920	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)))
92		ax-1cn 10330	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
93		fsumconst 14926	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
94	14, 92, 93	sylancl 580	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
95		hashfz1 13451	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
96	23, 95	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
97	96	oveq1d 6937	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((♯‘(1...(⌊‘𝐴))) · 1) = ((⌊‘𝐴) · 1))
98	82	mulid1d 10394	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) · 1) = (⌊‘𝐴))
99	94, 97, 98	3eqtrrd 2818	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))1)
100	91, 99	oveq12d 6940	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)) − Σ𝑑 ∈ (1...(⌊‘𝐴))1))
101	47	recnd 10405	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℂ)
102	14, 101, 84	fsumsub 14924	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)) − Σ𝑑 ∈ (1...(⌊‘𝐴))1))
103	100, 102	eqtr4d 2816	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) = Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1))
104		eqid 2777	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) = (ℤ_≥‘1)
105	104	uztrn2 12010	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)) → 𝑛 ∈ (ℤ_≥‘1))
106	105	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → 𝑛 ∈ (ℤ_≥‘1))
107	106	biantrurd 528	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
108		uzss 12013	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑑) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
109	108	ad2antll 719	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
110	109	sseld 3819	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
111	110	pm4.71rd 558	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((⌊‘𝐴) ∈ (ℤ_≥‘𝑛) ↔ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
112	107, 111	bitr3d 273	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → ((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ↔ ((⌊‘𝐴) ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
113	112	pm5.32da 574	. . . . . . . . 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)))))
114		ancom 454	. . . . . . . . 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))))
115		an4 646	. . . . . . . . 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))))
116	113, 114, 115	3bitr4g 306	. . . . . . . 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)))))
117		elfzuzb 12653	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
118		elfzuzb 12653	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...𝑛) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)))
119	117, 118	anbi12i 620	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ ((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))))
120		elfzuzb 12653	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
121		elfzuzb 12653	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑑...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
122	120, 121	anbi12i 620	. . . . . . . 8 ⊢ ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴))) ↔ ((𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)) ∧ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛))))
123	116, 119, 122	3bitr4g 306	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (𝑑...(⌊‘𝐴)))))
124	18	recnd 10405	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℂ)
125	124	anasss 460	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛))) → (1 / 𝑑) ∈ ℂ)
126	14, 14, 15, 123, 125	fsumcom2 14910	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
127	90, 103, 126	3brtr4d 4918	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
128	13, 36, 20, 44, 127	letrd 10533	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
129	27, 35	readdcld 10406	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ∈ ℝ)
130		elfznn 12687	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
131	130	adantl 475	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
132	131	nnrpd 12179	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
133	132	relogcld 24806	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
134		peano2re 10549	. . . . . . . 8 ⊢ ((log‘𝑛) ∈ ℝ → ((log‘𝑛) + 1) ∈ ℝ)
135	133, 134	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛) + 1) ∈ ℝ)
136		nnz 11751	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
137		flid 12928	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (⌊‘𝑛) = 𝑛)
138	136, 137	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (⌊‘𝑛) = 𝑛)
139	138	oveq2d 6938	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1...(⌊‘𝑛)) = (1...𝑛))
140	139	sumeq1d 14839	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) = Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
141		nnre 11382	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
142		nnge1 11404	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ≤ 𝑛)
143		harmonicubnd 25188	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ 1 ≤ 𝑛) → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
144	141, 142, 143	syl2anc 579	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
145	140, 144	eqbrtrrd 4910	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘𝑛) + 1))
146	131, 145	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘𝑛) + 1))
147	14, 19, 135, 146	fsumle 14935	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1))
148	133	recnd 10405	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℂ)
149		1cnd 10371	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
150	14, 148, 149	fsumadd 14877	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) + Σ𝑛 ∈ (1...(⌊‘𝐴))1))
151		logfac 24784	. . . . . . . . 9 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
152	23, 151	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
153		fsumconst 14926	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
154	14, 92, 153	sylancl 580	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
155	154, 97, 98	3eqtrrd 2818	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))1)
156	152, 155	oveq12d 6940	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) + Σ𝑛 ∈ (1...(⌊‘𝐴))1))
157	150, 156	eqtr4d 2816	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
158	147, 157	breqtrd 4912	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
159	35, 8, 27, 43	leadd2dd 10990	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
160	20, 129, 28, 158, 159	letrd 10533	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
161	13, 20, 28, 128, 160	letrd 10533	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
162	13, 8, 27	lesubaddd 10972	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) ≤ (log‘(!‘(⌊‘𝐴))) ↔ ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴)))
163	161, 162	mpbird 249	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) ≤ (log‘(!‘(⌊‘𝐴))))
164	12, 163	eqbrtrd 4908	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 Fincfn 8241 ℂcc 10270 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 < clt 10411 ≤ cle 10412 − cmin 10606 / cdiv 11032 ℕcn 11374 2c2 11430 ℕ₀cn0 11642 ℤcz 11728 ℤ_≥cuz 11992 ℝ⁺crp 12137 ...cfz 12643 ⌊cfl 12910 !cfa 13378 ♯chash 13435 Σcsu 14824 logclog 24738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-e 15201 df-sin 15202 df-cos 15203 df-pi 15205 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 df-em 25171

This theorem is referenced by: (None)

W3C validator