logfacbnd3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > logfacbnd3		Structured version Visualization version GIF version

Theorem logfacbnd3 27285

Description: Show the stronger statement log(𝑥!) = 𝑥log𝑥 − 𝑥 + 𝑂(log𝑥) alluded to in logfacrlim 27286. (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 13106	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
3		flge0nn0 13871	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
5	4	faccld 14333	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℕ)
6	5	nnrpd 13097	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
7		relogcl 26635	. . . . . . . 8 ⊢ ((!‘(⌊‘𝐴)) ∈ ℝ⁺ → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
9		rpre 13065	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
11		relogcl 26635	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
13		peano2rem 11603	. . . . . . . . 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 11320	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 · ((log‘𝐴) − 1)) ∈ ℝ)
16	8, 15	resubcld 11718	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℝ)
17	16	recnd 11318	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ)
18	17	abscld 15485	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ∈ ℝ)
19		peano2rem 11603	. . . 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 11242	. . . . 5 ⊢ 1 ∈ ℂ
22		subcl 11535	. . . . 5 ⊢ ((((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1) ∈ ℂ)
23	17, 21, 22	sylancl 585	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1) ∈ ℂ)
24	23	abscld 15485	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)) ∈ ℝ)
25		abs1 15346	. . . . 5 ⊢ (abs‘1) = 1
26	25	oveq2i 7459	. . . 4 ⊢ ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − (abs‘1)) = ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1)
27		abs2dif 15381	. . . . 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 585	. . . 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 5202	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)))
30		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
31	30	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
32	31	sumeq1d 15748	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
33		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
34		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
35	34	oveq1d 7463	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((log‘𝑥) − 1) = ((log‘𝐴) − 1))
36	33, 35	oveq12d 7466	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 · ((log‘𝑥) − 1)) = (𝐴 · ((log‘𝐴) − 1)))
37	32, 36	oveq12d 7466	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
38		eqid 2740	. . . . . . . . 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 7481	. . . . . . . . 9 ⊢ (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) ∈ V
40	37, 38, 39	fvmpt3i 7034	. . . . . . . 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 26661	. . . . . . . . 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 7463	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛) − (𝐴 · ((log‘𝐴) − 1))))
45	41, 44	eqtr4d 2783	. . . . . 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 13061	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
47		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
48		1z 12673	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
49		flid 13859	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → (⌊‘1) = 1)
50	48, 49	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (⌊‘1) = 1
51	47, 50	eqtrdi 2796	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (⌊‘𝑥) = 1)
52	51	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
53	52	sumeq1d 15748	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = Σ𝑛 ∈ (1...1)(log‘𝑛))
54		0cn 11282	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
55		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
56		log1 26645	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
57	55, 56	eqtrdi 2796	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
58	57	fsum1 15795	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)(log‘𝑛) = 0)
59	48, 54, 58	mp2an 691	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...1)(log‘𝑛) = 0
60	53, 59	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) = 0)
61		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → 𝑥 = 1)
62		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (log‘𝑥) = (log‘1))
63	62, 56	eqtrdi 2796	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (log‘𝑥) = 0)
64	63	oveq1d 7463	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((log‘𝑥) − 1) = (0 − 1))
65	61, 64	oveq12d 7466	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 · ((log‘𝑥) − 1)) = (1 · (0 − 1)))
66	54, 21	subcli 11612	. . . . . . . . . . . 12 ⊢ (0 − 1) ∈ ℂ
67	66	mullidi 11295	. . . . . . . . . . 11 ⊢ (1 · (0 − 1)) = (0 − 1)
68	65, 67	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑥 · ((log‘𝑥) − 1)) = (0 − 1))
69	60, 68	oveq12d 7466	. . . . . . . . 9 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = (0 − (0 − 1)))
70		nncan 11565	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → (0 − (0 − 1)) = 1)
71	54, 21, 70	mp2an 691	. . . . . . . . 9 ⊢ (0 − (0 − 1)) = 1
72	69, 71	eqtrdi 2796	. . . . . . . 8 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))(log‘𝑛) − (𝑥 · ((log‘𝑥) − 1))) = 1)
73	72, 38, 39	fvmpt3i 7034	. . . . . . 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 7466	. . . . 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 6924	. . . 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 13485	. . . . . 6 ⊢ (0(,)+∞) = ℝ⁺
78	77	eqcomi 2749	. . . . 5 ⊢ ℝ⁺ = (0(,)+∞)
79		nnuz 12946	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
80	48	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
81		1re 11290	. . . . . 6 ⊢ 1 ∈ ℝ
82	81	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
83		pnfxr 11344	. . . . . 6 ⊢ +∞ ∈ ℝ^*
84	83	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ^*)
85		1nn0 12569	. . . . . . 7 ⊢ 1 ∈ ℕ₀
86	81, 85	nn0addge1i 12601	. . . . . 6 ⊢ 1 ≤ (1 + 1)
87	86	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
88		0red 11293	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
89		rpre 13065	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
90	89	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
91		relogcl 26635	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
92	91	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
93		peano2rem 11603	. . . . . . 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 11320	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · ((log‘𝑥) − 1)) ∈ ℝ)
96		nnrp 13068	. . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
97	96, 92	sylan2 592	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → (log‘𝑥) ∈ ℝ)
98		advlog 26714	. . . . . 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 6920	. . . . 5 ⊢ (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
101		simp32 1210	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ∧ 𝑛 ≤ +∞)) → 𝑥 ≤ 𝑛)
102		logleb 26663	. . . . . . 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 26663	. . . . . . . 8 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
108	46, 106, 107	sylancr 586	. . . . . . 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 5202	. . . . 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 11918	. . . . . 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 11340	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ^*)
116		pnfge 13193	. . . . . 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 26095	. . . 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 5190	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (abs‘(((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1))) − 1)) ≤ (log‘𝐴))
120	20, 24, 12, 29, 119	letrd 11447	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) − 1) ≤ (log‘𝐴))
121	18, 82, 12	lesubaddd 11887	. 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 1537 ∈ wcel 2108 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 +∞cpnf 11321 ℝ^*cxr 11323 ≤ cle 11325 − cmin 11520 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ℝ⁺crp 13057 (,)cioo 13407 ...cfz 13567 ⌊cfl 13841 !cfa 14322 abscabs 15283 Σcsu 15734 D cdv 25918 logclog 26614

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616

This theorem is referenced by: logfacrlim 27286

W3C validator