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Theorem logfacbnd3 27191

Description: Show the stronger statement log(𝑥!) = 𝑥log𝑥 − 𝑥 + 𝑂(log𝑥) alluded to in logfacrlim 27192. (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 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
2	1	rprege0d 13063	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
3		flge0nn0 13842	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
5	4	faccld 14307	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℕ)
6	5	nnrpd 13054	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
7		relogcl 26541	. . . . . . . 8 ⊢ ((!‘(⌊‘𝐴)) ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
9		rpre 13022	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
11		relogcl 26541	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
13		peano2rem 11555	. . . . . . . . 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 11270	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 · ((log‘𝐴) − 1)) ∈ ℝ)
16	8, 15	resubcld 11670	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℝ)
17	16	recnd 11268	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ)
18	17	abscld 15460	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ∈ ℝ)
19		peano2rem 11555	. . . 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 11192	. . . . 5 ⊢ 1 ∈ ℂ
22		subcl 11486	. . . . 5 ⊢ ((((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1) ∈ ℂ)
23	17, 21, 22	sylancl 586	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1) ∈ ℂ)
24	23	abscld 15460	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)) ∈ ℝ)
25		abs1 15321	. . . . 5 ⊢ (abs‘1) = 1
26	25	oveq2i 7421	. . . 4 ⊢ ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − (abs‘1)) = ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1)
27		abs2dif 15356	. . . . 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 586	. . . 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 5160	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)))
30		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
31	30	oveq2d 7426	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
32	31	sumeq1d 15721	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
33		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
34		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
35	34	oveq1d 7425	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((log‘𝑥) − 1) = ((log‘𝐴) − 1))
36	33, 35	oveq12d 7428	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 · ((log‘𝑥) − 1)) = (𝐴 · ((log‘𝐴) − 1)))
37	32, 36	oveq12d 7428	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
38		eqid 2736	. . . . . . . . 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 7443	. . . . . . . . 9 ⊢ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) ∈ V
40	37, 38, 39	fvmpt3i 6996	. . . . . . . 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 480	. . . . . . 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 26567	. . . . . . . . 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 7425	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
45	41, 44	eqtr4d 2774	. . . . . 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 13017	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
47		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
48		1z 12627	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
49		flid 13830	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → (⌊‘1) = 1)
50	48, 49	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (⌊‘1) = 1
51	47, 50	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (⌊‘𝑥) = 1)
52	51	oveq2d 7426	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
53	52	sumeq1d 15721	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = Σ𝑛 ∈ (1...1)(log‘𝑛))
54		0cn 11232	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
55		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
56		log1 26551	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
57	55, 56	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
58	57	fsum1 15768	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)(log‘𝑛) = 0)
59	48, 54, 58	mp2an 692	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...1)(log‘𝑛) = 0
60	53, 59	eqtrdi 2787	. . . . . . . . . 10 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = 0)
61		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → 𝑥 = 1)
62		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (log‘𝑥) = (log‘1))
63	62, 56	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (log‘𝑥) = 0)
64	63	oveq1d 7425	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((log‘𝑥) − 1) = (0 − 1))
65	61, 64	oveq12d 7428	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 · ((log‘𝑥) − 1)) = (1 · (0 − 1)))
66	54, 21	subcli 11564	. . . . . . . . . . . 12 ⊢ (0 − 1) ∈ ℂ
67	66	mullidi 11245	. . . . . . . . . . 11 ⊢ (1 · (0 − 1)) = (0 − 1)
68	65, 67	eqtrdi 2787	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 · ((log‘𝑥) − 1)) = (0 − 1))
69	60, 68	oveq12d 7428	. . . . . . . . 9 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = (0 − (0 − 1)))
70		nncan 11517	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → (0 − (0 − 1)) = 1)
71	54, 21, 70	mp2an 692	. . . . . . . . 9 ⊢ (0 − (0 − 1)) = 1
72	69, 71	eqtrdi 2787	. . . . . . . 8 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = 1)
73	72, 38, 39	fvmpt3i 6996	. . . . . . 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 7428	. . . . 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 6885	. . . 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 13447	. . . . . 6 ⊢ (0(,)+∞) = ℝ⁺
78	77	eqcomi 2745	. . . . 5 ⊢ ℝ⁺ = (0(,)+∞)
79		nnuz 12900	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
80	48	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
81		1re 11240	. . . . . 6 ⊢ 1 ∈ ℝ
82	81	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
83		pnfxr 11294	. . . . . 6 ⊢ +∞ ∈ ℝ^*
84	83	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ^*)
85		1nn0 12522	. . . . . . 7 ⊢ 1 ∈ ℕ₀
86	81, 85	nn0addge1i 12554	. . . . . 6 ⊢ 1 ≤ (1 + 1)
87	86	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
88		0red 11243	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
89		rpre 13022	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
90	89	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
91		relogcl 26541	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
92	91	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
93		peano2rem 11555	. . . . . . 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 11270	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · ((log‘𝑥) − 1)) ∈ ℝ)
96		nnrp 13025	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
97	96, 92	sylan2 593	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → (log‘𝑥) ∈ ℝ)
98		advlog 26620	. . . . . 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 6881	. . . . 5 ⊢ (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
101		simp32 1211	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ≤ 𝑛)
102		logleb 26569	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 ≤ 𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
103	102	3ad2ant2 1134	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (𝑥 ≤ 𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
104	101, 103	mpbid 232	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → (log‘𝑥) ≤ (log‘𝑛))
105		simprr 772	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
106		simprl 770	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
107		logleb 26569	. . . . . . . 8 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
108	46, 106, 107	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
109	105, 108	mpbid 232	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
110	56, 109	eqbrtrrid 5160	. . . . 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 11870	. . . . . 6 ⊢ 1 ≤ 1
113	112	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
114		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
115	10	rexrd 11290	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ^*)
116		pnfge 13151	. . . . . 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 25998	. . . 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 5148	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)) ≤ (log‘𝐴))
120	20, 24, 12, 29, 119	letrd 11397	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (log‘𝐴))
121	18, 82, 12	lesubaddd 11839	. 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 232	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5124 ↦ cmpt 5206 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 +∞cpnf 11271 ℝ^*cxr 11273 ≤ cle 11275 − cmin 11471 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ℝ⁺crp 13013 (,)cioo 13367 ...cfz 13529 ⌊cfl 13812 !cfa 14296 abscabs 15258 Σcsu 15707 D cdv 25821 logclog 26520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-pi 16093 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825 df-log 26522

This theorem is referenced by: logfacrlim 27192

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