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Theorem logfaclbnd 27110

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 12990	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
2	1	times2d 12460	. . . 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 26464	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 11246	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6		2cnd 12294	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
7	1, 5, 6	subdid 11674	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = ((𝐴 · (log‘𝐴)) − (𝐴 · 2)))
8		rpre 12988	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
9	8, 4	remulcld 11248	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℝ)
10	9	recnd 11246	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · (log‘𝐴)) ∈ ℂ)
11	10, 1, 1	subsub4d 11606	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴) = ((𝐴 · (log‘𝐴)) − (𝐴 + 𝐴)))
12	3, 7, 11	3eqtr4d 2776	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) = (((𝐴 · (log‘𝐴)) − 𝐴) − 𝐴))
13	9, 8	resubcld 11646	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ∈ ℝ)
14		fzfid 13944	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
15		fzfid 13944	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
16		elfznn 13536	. . . . . . . 8 ⊢ (𝑑 ∈ (1...𝑛) → 𝑑 ∈ ℕ)
17	16	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → 𝑑 ∈ ℕ)
18	17	nnrecred 12267	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ (1...𝑛)) → (1 / 𝑑) ∈ ℝ)
19	15, 18	fsumrecl 15686	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
20	14, 19	fsumrecl 15686	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ∈ ℝ)
21		rprege0 12995	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
22		flge0nn0 13791	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
24	23	faccld 14249	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℕ)
25	24	nnrpd 13020	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
26	25	relogcld 26512	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
27	26, 8	readdcld 11247	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + 𝐴) ∈ ℝ)
28		elfznn 13536	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℕ)
29	28	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
30	29	nnrecred 12267	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ)
31	14, 30	fsumrecl 15686	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑) ∈ ℝ)
32	8, 31	remulcld 11248	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) ∈ ℝ)
33		reflcl 13767	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
34	8, 33	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
35	32, 34	resubcld 11646	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) ∈ ℝ)
36		harmoniclbnd 26896	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑))
37		rpregt0 12994	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
38		lemul2 12071	. . . . . . . 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 1368	. . . . . . 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 13770	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
42	8, 41	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
43	9, 34, 32, 8, 40, 42	le2subd 11838	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)))
44	28	nnrecred 12267	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (1 / 𝑑) ∈ ℝ)
45		remulcl 11197	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 𝑑) ∈ ℝ) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
46	8, 44, 45	syl2an 595	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℝ)
47		peano2rem 11531	. . . . . . . 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 13944	. . . . . . . 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 15686	. . . . . . 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 11391	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ ℝ → ((⌊‘𝐴) + 1) ∈ ℝ)
55	53, 54	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((⌊‘𝐴) + 1) ∈ ℝ)
56	29	nnred 12231	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ)
57		fllep1 13772	. . . . . . . . . . . 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 11834	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ≤ (((⌊‘𝐴) + 1) − 𝑑))
61	52, 56	resubcld 11646	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 − 𝑑) ∈ ℝ)
62	55, 56	resubcld 11646	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) ∈ ℝ)
63	29	nnrpd 13020	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ⁺)
64	63	rpreccld 13032	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℝ⁺)
65	61, 62, 64	lemul1d 13065	. . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℂ)
68	29	nncnd 12232	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℂ)
69	30	recnd 11246	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1 / 𝑑) ∈ ℂ)
70	67, 68, 69	subdird 11675	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 − 𝑑) · (1 / 𝑑)) = ((𝐴 · (1 / 𝑑)) − (𝑑 · (1 / 𝑑))))
71	29	nnne0d 12266	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ≠ 0)
72	68, 71	recidd 11989	. . . . . . . . . 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 2767	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) = ((𝐴 − 𝑑) · (1 / 𝑑)))
75		fsumconst 15742	. . . . . . . . . 10 ⊢ (((𝑑...(⌊‘𝐴)) ∈ Fin ∧ (1 / 𝑑) ∈ ℂ) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
76	49, 69, 75	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((♯‘(𝑑...(⌊‘𝐴))) · (1 / 𝑑)))
77		elfzuz3 13504	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
78	77	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ (ℤ_≥‘𝑑))
79		hashfz 14392	. . . . . . . . . . . 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 11246	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
82	81	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (⌊‘𝐴) ∈ ℂ)
83		1cnd 11213	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
84	82, 83, 68	addsubd 11596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((⌊‘𝐴) + 1) − 𝑑) = (((⌊‘𝐴) − 𝑑) + 1))
85	80, 84	eqtr4d 2769	. . . . . . . . . 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 2766	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑) = ((((⌊‘𝐴) + 1) − 𝑑) · (1 / 𝑑)))
88	66, 74, 87	3brtr4d 5173	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
89	14, 48, 51, 88	fsumle 15751	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) ≤ Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
90	14, 1, 69	fsummulc2 15736	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)))
91		ax-1cn 11170	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
92		fsumconst 15742	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
93	14, 91, 92	sylancl 585	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
94		hashfz1 14311	. . . . . . . . . . 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 11235	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) · 1) = (⌊‘𝐴))
98	93, 96, 97	3eqtrrd 2771	. . . . . . . 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 11246	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 · (1 / 𝑑)) ∈ ℂ)
101	14, 100, 83	fsumsub 15740	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1) = (Σ𝑑 ∈ (1...(⌊‘𝐴))(𝐴 · (1 / 𝑑)) − Σ𝑑 ∈ (1...(⌊‘𝐴))1))
102	99, 101	eqtr4d 2769	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · Σ𝑑 ∈ (1...(⌊‘𝐴))(1 / 𝑑)) − (⌊‘𝐴)) = Σ𝑑 ∈ (1...(⌊‘𝐴))((𝐴 · (1 / 𝑑)) − 1))
103		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) = (ℤ_≥‘1)
104	103	uztrn2 12845	. . . . . . . . . . . . 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 12849	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑑) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
108	107	ad2antll 726	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑑))
109	108	sseld 3976	. . . . . . . . . . . 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 578	. . . . . . . . 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 653	. . . . . . . . 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 13501	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
117		elfzuzb 13501	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...𝑛) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑)))
118	116, 117	anbi12i 626	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ (1...𝑛)) ↔ ((𝑛 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)) ∧ (𝑑 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑑))))
119		elfzuzb 13501	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑑)))
120		elfzuzb 13501	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑑...(⌊‘𝐴)) ↔ (𝑛 ∈ (ℤ_≥‘𝑑) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘𝑛)))
121	119, 120	anbi12i 626	. . . . . . . 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 11246	. . . . . . . 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 15726	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (𝑑...(⌊‘𝐴))(1 / 𝑑))
126	89, 102, 125	3brtr4d 5173	. . . . 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 11375	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
128	26, 34	readdcld 11247	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ∈ ℝ)
129		elfznn 13536	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
130	129	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
131	130	nnrpd 13020	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
132	131	relogcld 26512	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
133		peano2re 11391	. . . . . . . 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 12583	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
136		flid 13779	. . . . . . . . . . . 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 15653	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) = Σ𝑑 ∈ (1...𝑛)(1 / 𝑑))
140		nnre 12223	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
141		nnge1 12244	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ≤ 𝑛)
142		harmonicubnd 26897	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ 1 ≤ 𝑛) → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
143	140, 141, 142	syl2anc 583	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ (1...(⌊‘𝑛))(1 / 𝑑) ≤ ((log‘𝑛) + 1))
144	139, 143	eqbrtrrd 5165	. . . . . . . 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 15751	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1))
147	132	recnd 11246	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℂ)
148		1cnd 11213	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℂ)
149	14, 147, 148	fsumadd 15692	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) + Σ𝑛 ∈ (1...(⌊‘𝐴))1))
150		logfac 26490	. . . . . . . . 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 15742	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
153	14, 91, 152	sylancl 585	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))1 = ((♯‘(1...(⌊‘𝐴))) · 1))
154	153, 96, 97	3eqtrrd 2771	. . . . . . . 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 2769	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛) + 1) = ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
157	146, 156	breqtrd 5167	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)))
158	34, 8, 26, 42	leadd2dd 11833	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝐴))) + (⌊‘𝐴)) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
159	20, 128, 27, 157, 158	letrd 11375	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ (1...𝑛)(1 / 𝑑) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
160	13, 20, 27, 127, 159	letrd 11375	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 · (log‘𝐴)) − 𝐴) ≤ ((log‘(!‘(⌊‘𝐴))) + 𝐴))
161	13, 8, 26	lesubaddd 11815	. . 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 5163	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 ℕcn 12216 2c2 12271 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12980 ...cfz 13490 ⌊cfl 13761 !cfa 14238 ♯chash 14295 Σcsu 15638 logclog 26443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-e 16018 df-sin 16019 df-cos 16020 df-tan 16021 df-pi 16022 df-dvds 16205 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-cmp 23246 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751 df-ulm 26268 df-log 26445 df-atan 26754 df-em 26880

This theorem is referenced by: (None)

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