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Theorem logfacbnd3 27205

Description: Show the stronger statement log(𝑥!) = 𝑥log𝑥 − 𝑥 + 𝑂(log𝑥) alluded to in logfacrlim 27206. (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 12982	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
3		flge0nn0 13768	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
5	4	faccld 14235	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℕ)
6	5	nnrpd 12973	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
7		relogcl 26555	. . . . . . . 8 ⊢ ((!‘(⌊‘𝐴)) ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
9		rpre 12940	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
11		relogcl 26555	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
13		peano2rem 11450	. . . . . . . . 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 11164	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 · ((log‘𝐴) − 1)) ∈ ℝ)
16	8, 15	resubcld 11567	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℝ)
17	16	recnd 11162	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ)
18	17	abscld 15390	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ∈ ℝ)
19		peano2rem 11450	. . . 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 11085	. . . . 5 ⊢ 1 ∈ ℂ
22		subcl 11381	. . . . 5 ⊢ ((((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1) ∈ ℂ)
23	17, 21, 22	sylancl 587	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1) ∈ ℂ)
24	23	abscld 15390	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)) ∈ ℝ)
25		abs1 15248	. . . . 5 ⊢ (abs‘1) = 1
26	25	oveq2i 7369	. . . 4 ⊢ ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − (abs‘1)) = ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1)
27		abs2dif 15284	. . . . 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 587	. . . 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 5122	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)))
30		fveq2 6832	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
31	30	oveq2d 7374	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
32	31	sumeq1d 15651	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
33		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
34		fveq2 6832	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
35	34	oveq1d 7373	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((log‘𝑥) − 1) = ((log‘𝐴) − 1))
36	33, 35	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 · ((log‘𝑥) − 1)) = (𝐴 · ((log‘𝐴) − 1)))
37	32, 36	oveq12d 7376	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
38		eqid 2737	. . . . . . . . 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 7391	. . . . . . . . 9 ⊢ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) ∈ V
40	37, 38, 39	fvmpt3i 6945	. . . . . . . 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 26581	. . . . . . . . 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 7373	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
45	41, 44	eqtr4d 2775	. . . . . 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 12935	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
47		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
48		1z 12546	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
49		flid 13756	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → (⌊‘1) = 1)
50	48, 49	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (⌊‘1) = 1
51	47, 50	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (⌊‘𝑥) = 1)
52	51	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
53	52	sumeq1d 15651	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = Σ𝑛 ∈ (1...1)(log‘𝑛))
54		0cn 11125	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
55		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
56		log1 26565	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
57	55, 56	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
58	57	fsum1 15698	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)(log‘𝑛) = 0)
59	48, 54, 58	mp2an 693	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...1)(log‘𝑛) = 0
60	53, 59	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = 0)
61		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → 𝑥 = 1)
62		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (log‘𝑥) = (log‘1))
63	62, 56	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (log‘𝑥) = 0)
64	63	oveq1d 7373	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((log‘𝑥) − 1) = (0 − 1))
65	61, 64	oveq12d 7376	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 · ((log‘𝑥) − 1)) = (1 · (0 − 1)))
66	54, 21	subcli 11459	. . . . . . . . . . . 12 ⊢ (0 − 1) ∈ ℂ
67	66	mullidi 11139	. . . . . . . . . . 11 ⊢ (1 · (0 − 1)) = (0 − 1)
68	65, 67	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 · ((log‘𝑥) − 1)) = (0 − 1))
69	60, 68	oveq12d 7376	. . . . . . . . 9 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = (0 − (0 − 1)))
70		nncan 11412	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → (0 − (0 − 1)) = 1)
71	54, 21, 70	mp2an 693	. . . . . . . . 9 ⊢ (0 − (0 − 1)) = 1
72	69, 71	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = 1)
73	72, 38, 39	fvmpt3i 6945	. . . . . . 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 7376	. . . . 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 6836	. . . 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 13367	. . . . . 6 ⊢ (0(,)+∞) = ℝ⁺
78	77	eqcomi 2746	. . . . 5 ⊢ ℝ⁺ = (0(,)+∞)
79		nnuz 12816	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
80	48	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
81		1re 11133	. . . . . 6 ⊢ 1 ∈ ℝ
82	81	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
83		pnfxr 11188	. . . . . 6 ⊢ +∞ ∈ ℝ^*
84	83	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ^*)
85		1nn0 12442	. . . . . . 7 ⊢ 1 ∈ ℕ₀
86	81, 85	nn0addge1i 12474	. . . . . 6 ⊢ 1 ≤ (1 + 1)
87	86	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
88		0red 11136	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
89		rpre 12940	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
90	89	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
91		relogcl 26555	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
92	91	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
93		peano2rem 11450	. . . . . . 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 11164	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · ((log‘𝑥) − 1)) ∈ ℝ)
96		nnrp 12943	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
97	96, 92	sylan2 594	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → (log‘𝑥) ∈ ℝ)
98		advlog 26634	. . . . . 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 6832	. . . . 5 ⊢ (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
101		simp32 1212	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ≤ 𝑛)
102		logleb 26583	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 ≤ 𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
103	102	3ad2ant2 1135	. . . . . 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 773	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
106		simprl 771	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
107		logleb 26583	. . . . . . . 8 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
108	46, 106, 107	sylancr 588	. . . . . . 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 5122	. . . . 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 11767	. . . . . 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 11184	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ^*)
116		pnfge 13070	. . . . . 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 26013	. . . 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 5110	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)) ≤ (log‘𝐴))
120	20, 24, 12, 29, 119	letrd 11292	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (log‘𝐴))
121	18, 82, 12	lesubaddd 11736	. 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 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6490 (class class class)co 7358 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 +∞cpnf 11165 ℝ^*cxr 11167 ≤ cle 11169 − cmin 11366 ℕcn 12163 ℕ₀cn0 12426 ℤcz 12513 ℝ⁺crp 12931 (,)cioo 13287 ...cfz 13450 ⌊cfl 13738 !cfa 14224 abscabs 15185 Σcsu 15637 D cdv 25839 logclog 26534

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

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 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21334 df-xmet 21335 df-met 21336 df-bl 21337 df-mopn 21338 df-fbas 21339 df-fg 21340 df-cnfld 21343 df-top 22868 df-topon 22885 df-topsp 22907 df-bases 22920 df-cld 22993 df-ntr 22994 df-cls 22995 df-nei 23072 df-lp 23110 df-perf 23111 df-cn 23201 df-cnp 23202 df-haus 23289 df-cmp 23361 df-tx 23536 df-hmeo 23729 df-fil 23820 df-fm 23912 df-flim 23913 df-flf 23914 df-xms 24294 df-ms 24295 df-tms 24296 df-cncf 24854 df-limc 25842 df-dv 25843 df-log 26536

This theorem is referenced by: logfacrlim 27206

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