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Theorem logfacbnd3 25799

Description: Show the stronger statement log(𝑥!) = 𝑥log𝑥 − 𝑥 + 𝑂(log𝑥) alluded to in logfacrlim 25800. (Contributed by Mario Carneiro, 20-May-2016.)

**Assertion**
Ref	Expression
logfacbnd3	⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ≤ ((log‘𝐴) + 1))

Proof of Theorem logfacbnd3 Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
2	1	rprege0d 12439	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
3		flge0nn0 13191	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
5	4	faccld 13645	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℕ)
6	5	nnrpd 12430	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
7		relogcl 25159	. . . . . . . 8 ⊢ ((!‘(⌊‘𝐴)) ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
9		rpre 12398	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
10	9	adantr 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
11		relogcl 25159	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
12	11	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
13		peano2rem 10953	. . . . . . . . 9 ⊢ ((log‘𝐴) ∈ ℝ → ((log‘𝐴) − 1) ∈ ℝ)
14	12, 13	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘𝐴) − 1) ∈ ℝ)
15	10, 14	remulcld 10671	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 · ((log‘𝐴) − 1)) ∈ ℝ)
16	8, 15	resubcld 11068	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℝ)
17	16	recnd 10669	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ)
18	17	abscld 14796	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ∈ ℝ)
19		peano2rem 10953	. . . 4 ⊢ ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ∈ ℝ → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ∈ ℝ)
20	18, 19	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ∈ ℝ)
21		ax-1cn 10595	. . . . 5 ⊢ 1 ∈ ℂ
22		subcl 10885	. . . . 5 ⊢ ((((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1) ∈ ℂ)
23	17, 21, 22	sylancl 588	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1) ∈ ℂ)
24	23	abscld 14796	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)) ∈ ℝ)
25		abs1 14657	. . . . 5 ⊢ (abs‘1) = 1
26	25	oveq2i 7167	. . . 4 ⊢ ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − (abs‘1)) = ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1)
27		abs2dif 14692	. . . . 5 ⊢ ((((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ ∧ 1 ∈ ℂ) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − (abs‘1)) ≤ (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)))
28	17, 21, 27	sylancl 588	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − (abs‘1)) ≤ (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)))
29	26, 28	eqbrtrrid 5102	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)))
30		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
31	30	oveq2d 7172	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
32	31	sumeq1d 15058	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
33		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
34		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
35	34	oveq1d 7171	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((log‘𝑥) − 1) = ((log‘𝐴) − 1))
36	33, 35	oveq12d 7174	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 · ((log‘𝑥) − 1)) = (𝐴 · ((log‘𝐴) − 1)))
37	32, 36	oveq12d 7174	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
38		eqid 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))
39		ovex 7189	. . . . . . . . 9 ⊢ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) ∈ V
40	37, 38, 39	fvmpt3i 6773	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
41	40	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
42		logfac 25184	. . . . . . . . 9 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
43	4, 42	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
44	43	oveq1d 7171	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
45	41, 44	eqtr4d 2859	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘𝐴) = ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))))
46		1rp 12394	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
47		fveq2 6670	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
48		1z 12013	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
49		flid 13179	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → (⌊‘1) = 1)
50	48, 49	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (⌊‘1) = 1
51	47, 50	syl6eq 2872	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (⌊‘𝑥) = 1)
52	51	oveq2d 7172	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
53	52	sumeq1d 15058	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = Σ𝑛 ∈ (1...1)(log‘𝑛))
54		0cn 10633	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
55		fveq2 6670	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
56		log1 25169	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
57	55, 56	syl6eq 2872	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
58	57	fsum1 15102	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)(log‘𝑛) = 0)
59	48, 54, 58	mp2an 690	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...1)(log‘𝑛) = 0
60	53, 59	syl6eq 2872	. . . . . . . . . 10 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = 0)
61		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → 𝑥 = 1)
62		fveq2 6670	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (log‘𝑥) = (log‘1))
63	62, 56	syl6eq 2872	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (log‘𝑥) = 0)
64	63	oveq1d 7171	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((log‘𝑥) − 1) = (0 − 1))
65	61, 64	oveq12d 7174	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 · ((log‘𝑥) − 1)) = (1 · (0 − 1)))
66	54, 21	subcli 10962	. . . . . . . . . . . 12 ⊢ (0 − 1) ∈ ℂ
67	66	mulid2i 10646	. . . . . . . . . . 11 ⊢ (1 · (0 − 1)) = (0 − 1)
68	65, 67	syl6eq 2872	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 · ((log‘𝑥) − 1)) = (0 − 1))
69	60, 68	oveq12d 7174	. . . . . . . . 9 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = (0 − (0 − 1)))
70		nncan 10915	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → (0 − (0 − 1)) = 1)
71	54, 21, 70	mp2an 690	. . . . . . . . 9 ⊢ (0 − (0 − 1)) = 1
72	69, 71	syl6eq 2872	. . . . . . . 8 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = 1)
73	72, 38, 39	fvmpt3i 6773	. . . . . . 7 ⊢ (1 ∈ ℝ⁺ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘1) = 1)
74	46, 73	mp1i 13	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘1) = 1)
75	45, 74	oveq12d 7174	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘𝐴) − ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘1)) = (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1))
76	75	fveq2d 6674	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘𝐴) − ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘1))) = (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)))
77		ioorp 12815	. . . . . 6 ⊢ (0(,)+∞) = ℝ⁺
78	77	eqcomi 2830	. . . . 5 ⊢ ℝ⁺ = (0(,)+∞)
79		nnuz 12282	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
80	48	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
81		1re 10641	. . . . . 6 ⊢ 1 ∈ ℝ
82	81	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
83		pnfxr 10695	. . . . . 6 ⊢ +∞ ∈ ℝ^*
84	83	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ^*)
85		1nn0 11914	. . . . . . 7 ⊢ 1 ∈ ℕ₀
86	81, 85	nn0addge1i 11946	. . . . . 6 ⊢ 1 ≤ (1 + 1)
87	86	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
88		0red 10644	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
89		rpre 12398	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
90	89	adantl 484	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
91		relogcl 25159	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
92	91	adantl 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
93		peano2rem 10953	. . . . . . 7 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ)
94	92, 93	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℝ)
95	90, 94	remulcld 10671	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · ((log‘𝑥) − 1)) ∈ ℝ)
96		nnrp 12401	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
97	96, 92	sylan2 594	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → (log‘𝑥) ∈ ℝ)
98		advlog 25237	. . . . . 6 ⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
99	98	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
100		fveq2 6670	. . . . 5 ⊢ (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
101		simp32 1206	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ≤ 𝑛)
102		logleb 25186	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 ≤ 𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
103	102	3ad2ant2 1130	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (𝑥 ≤ 𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
104	101, 103	mpbid 234	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘𝑥) ≤ (log‘𝑛))
105		simprr 771	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
106		simprl 769	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
107		logleb 25186	. . . . . . . 8 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
108	46, 106, 107	sylancr 589	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
109	105, 108	mpbid 234	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
110	56, 109	eqbrtrrid 5102	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
111	46	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ⁺)
112		1le1 11268	. . . . . 6 ⊢ 1 ≤ 1
113	112	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
114		simpr 487	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
115	10	rexrd 10691	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ^*)
116		pnfge 12526	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
117	115, 116	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ≤ +∞)
118	78, 79, 80, 82, 84, 87, 88, 95, 92, 97, 99, 100, 104, 38, 110, 111, 1, 113, 114, 117, 34	dvfsum2 24631	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘𝐴) − ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))))‘1))) ≤ (log‘𝐴))
119	76, 118	eqbrtrrd 5090	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)) ≤ (log‘𝐴))
120	20, 24, 12, 29, 119	letrd 10797	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (log‘𝐴))
121	18, 82, 12	lesubaddd 11237	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (log‘𝐴) ↔ (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ≤ ((log‘𝐴) + 1)))
122	120, 121	mpbid 234	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ≤ ((log‘𝐴) + 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 +∞cpnf 10672 ℝ^*cxr 10674 ≤ cle 10676 − cmin 10870 ℕcn 11638 ℕ₀cn0 11898 ℤcz 11982 ℝ⁺crp 12390 (,)cioo 12739 ...cfz 12893 ⌊cfl 13161 !cfa 13634 abscabs 14593 Σcsu 15042 D cdv 24461 logclog 25138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140

This theorem is referenced by: logfacrlim 25800

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