logfacrlim - Metamath Proof Explorer

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

Theorem logfacrlim 27283

Description: Combine the estimates logfacubnd 27280 and logfaclbnd 27281, to get log(𝑥!) = 𝑥log𝑥 + 𝑂(𝑥). Equation 9.2.9 of [Shapiro], p. 329. This is a weak form of the even stronger statement, log(𝑥!) = 𝑥log𝑥 − 𝑥 + 𝑂(log𝑥). (Contributed by Mario Carneiro, 16-Apr-2016.) (Revised by Mario Carneiro, 21-May-2016.)

**Assertion**
Ref	Expression
logfacrlim	⊢ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝_𝑟 1

**Proof of Theorem logfacrlim**
Step	Hyp	Ref	Expression
1		1red 11260	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
2		1cnd 11254	. . 3 ⊢ (⊤ → 1 ∈ ℂ)
3		relogcl 26632	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
4	3	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
5	4	recnd 11287	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
6		1cnd 11254	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
7		rpcnne0 13051	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
8	7	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
9		divdir 11945	. . . . . . 7 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((log‘𝑥) + 1) / 𝑥) = (((log‘𝑥) / 𝑥) + (1 / 𝑥)))
10	5, 6, 8, 9	syl3anc 1370	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) + 1) / 𝑥) = (((log‘𝑥) / 𝑥) + (1 / 𝑥)))
11	10	mpteq2dva 5248	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((log‘𝑥) + 1) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((log‘𝑥) / 𝑥) + (1 / 𝑥))))
12		simpr 484	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
13	4, 12	rerpdivcld 13106	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / 𝑥) ∈ ℝ)
14		rpreccl 13059	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
15	14	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
16	15	rpred 13075	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
17	8	simpld 494	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
18	17	cxp1d 26763	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐1) = 𝑥)
19	18	oveq2d 7447	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (𝑥↑_𝑐1)) = ((log‘𝑥) / 𝑥))
20	19	mpteq2dva 5248	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (𝑥↑_𝑐1))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 𝑥)))
21		1rp 13036	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
22		cxploglim 27036	. . . . . . . 8 ⊢ (1 ∈ ℝ⁺ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (𝑥↑_𝑐1))) ⇝_𝑟 0)
23	21, 22	mp1i 13	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (𝑥↑_𝑐1))) ⇝_𝑟 0)
24	20, 23	eqbrtrrd 5172	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 𝑥)) ⇝_𝑟 0)
25		ax-1cn 11211	. . . . . . 7 ⊢ 1 ∈ ℂ
26		divrcnv 15885	. . . . . . 7 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
27	25, 26	mp1i 13	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
28	13, 16, 24, 27	rlimadd 15676	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((log‘𝑥) / 𝑥) + (1 / 𝑥))) ⇝_𝑟 (0 + 0))
29	11, 28	eqbrtrd 5170	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((log‘𝑥) + 1) / 𝑥)) ⇝_𝑟 (0 + 0))
30		00id 11434	. . . 4 ⊢ (0 + 0) = 0
31	29, 30	breqtrdi 5189	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((log‘𝑥) + 1) / 𝑥)) ⇝_𝑟 0)
32		peano2re 11432	. . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) + 1) ∈ ℝ)
33	4, 32	syl 17	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) + 1) ∈ ℝ)
34	33, 12	rerpdivcld 13106	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) + 1) / 𝑥) ∈ ℝ)
35	34	recnd 11287	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) + 1) / 𝑥) ∈ ℂ)
36		rprege0 13048	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
37	36	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
38		flge0nn0 13857	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
39		faccl 14319	. . . . . . . . 9 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (!‘(⌊‘𝑥)) ∈ ℕ)
40	37, 38, 39	3syl 18	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (!‘(⌊‘𝑥)) ∈ ℕ)
41	40	nnrpd 13073	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (!‘(⌊‘𝑥)) ∈ ℝ⁺)
42		relogcl 26632	. . . . . . 7 ⊢ ((!‘(⌊‘𝑥)) ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℝ)
43	41, 42	syl 17	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘(!‘(⌊‘𝑥))) ∈ ℝ)
44	43, 12	rerpdivcld 13106	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
45	44	recnd 11287	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℂ)
46	5, 45	subcld 11618	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ ℂ)
47		logfacbnd3 27282	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) ≤ ((log‘𝑥) + 1))
48	47	adantl 481	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) ≤ ((log‘𝑥) + 1))
49	43	recnd 11287	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘(!‘(⌊‘𝑥))) ∈ ℂ)
50	49	adantrr 717	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘(!‘(⌊‘𝑥))) ∈ ℂ)
51	7	ad2antrl 728	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
52	51	simpld 494	. . . . . . . . 9 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℂ)
53	5	adantrr 717	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℂ)
54		subcl 11505	. . . . . . . . . 10 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → ((log‘𝑥) − 1) ∈ ℂ)
55	53, 25, 54	sylancl 586	. . . . . . . . 9 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) − 1) ∈ ℂ)
56	52, 55	mulcld 11279	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · ((log‘𝑥) − 1)) ∈ ℂ)
57	50, 56	subcld 11618	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1))) ∈ ℂ)
58	57	abscld 15472	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) ∈ ℝ)
59	4	adantrr 717	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
60	59, 32	syl 17	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + 1) ∈ ℝ)
61		rpregt0 13047	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
62	61	ad2antrl 728	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
63		lediv1 12131	. . . . . 6 ⊢ (((abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) ∈ ℝ ∧ ((log‘𝑥) + 1) ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → ((abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) ≤ ((log‘𝑥) + 1) ↔ ((abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) / 𝑥) ≤ (((log‘𝑥) + 1) / 𝑥)))
64	58, 60, 62, 63	syl3anc 1370	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) ≤ ((log‘𝑥) + 1) ↔ ((abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) / 𝑥) ≤ (((log‘𝑥) + 1) / 𝑥)))
65	48, 64	mpbid 232	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) / 𝑥) ≤ (((log‘𝑥) + 1) / 𝑥))
66	51	simprd 495	. . . . . . . . 9 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ≠ 0)
67	55, 52, 66	divcan3d 12046	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑥 · ((log‘𝑥) − 1)) / 𝑥) = ((log‘𝑥) − 1))
68	67	oveq1d 7446	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((𝑥 · ((log‘𝑥) − 1)) / 𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) = (((log‘𝑥) − 1) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))
69		divsubdir 11959	. . . . . . . 8 ⊢ (((𝑥 · ((log‘𝑥) − 1)) ∈ ℂ ∧ (log‘(!‘(⌊‘𝑥))) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) = (((𝑥 · ((log‘𝑥) − 1)) / 𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))
70	56, 50, 51, 69	syl3anc 1370	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) = (((𝑥 · ((log‘𝑥) − 1)) / 𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))
71	45	adantrr 717	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℂ)
72		1cnd 11254	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ ℂ)
73	53, 71, 72	sub32d 11650	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − 1) = (((log‘𝑥) − 1) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))
74	68, 70, 73	3eqtr4rd 2786	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − 1) = (((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
75	74	fveq2d 6911	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − 1)) = (abs‘(((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥)))) / 𝑥)))
76	56, 50	subcld 11618	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥)))) ∈ ℂ)
77	76, 52, 66	absdivd 15491	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥)))) / 𝑥)) = ((abs‘((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥))))) / (abs‘𝑥)))
78	56, 50	abssubd 15489	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥))))) = (abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))))
79	36	ad2antrl 728	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
80		absid 15332	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
81	79, 80	syl 17	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘𝑥) = 𝑥)
82	78, 81	oveq12d 7449	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((abs‘((𝑥 · ((log‘𝑥) − 1)) − (log‘(!‘(⌊‘𝑥))))) / (abs‘𝑥)) = ((abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) / 𝑥))
83	75, 77, 82	3eqtrd 2779	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − 1)) = ((abs‘((log‘(!‘(⌊‘𝑥))) − (𝑥 · ((log‘𝑥) − 1)))) / 𝑥))
84	35	adantrr 717	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((log‘𝑥) + 1) / 𝑥) ∈ ℂ)
85	84	subid1d 11607	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((((log‘𝑥) + 1) / 𝑥) − 0) = (((log‘𝑥) + 1) / 𝑥))
86	85	fveq2d 6911	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((((log‘𝑥) + 1) / 𝑥) − 0)) = (abs‘(((log‘𝑥) + 1) / 𝑥)))
87		log1 26642	. . . . . . . . 9 ⊢ (log‘1) = 0
88		simprr 773	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
89	12	adantrr 717	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
90		logleb 26660	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
91	21, 89, 90	sylancr 587	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
92	88, 91	mpbid 232	. . . . . . . . 9 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
93	87, 92	eqbrtrrid 5184	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
94	59, 93	ge0p1rpd 13105	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + 1) ∈ ℝ⁺)
95	94, 89	rpdivcld 13092	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((log‘𝑥) + 1) / 𝑥) ∈ ℝ⁺)
96		rprege0 13048	. . . . . 6 ⊢ ((((log‘𝑥) + 1) / 𝑥) ∈ ℝ⁺ → ((((log‘𝑥) + 1) / 𝑥) ∈ ℝ ∧ 0 ≤ (((log‘𝑥) + 1) / 𝑥)))
97		absid 15332	. . . . . 6 ⊢ (((((log‘𝑥) + 1) / 𝑥) ∈ ℝ ∧ 0 ≤ (((log‘𝑥) + 1) / 𝑥)) → (abs‘(((log‘𝑥) + 1) / 𝑥)) = (((log‘𝑥) + 1) / 𝑥))
98	95, 96, 97	3syl 18	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((log‘𝑥) + 1) / 𝑥)) = (((log‘𝑥) + 1) / 𝑥))
99	86, 98	eqtrd 2775	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((((log‘𝑥) + 1) / 𝑥) − 0)) = (((log‘𝑥) + 1) / 𝑥))
100	65, 83, 99	3brtr4d 5180	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − 1)) ≤ (abs‘((((log‘𝑥) + 1) / 𝑥) − 0)))
101	1, 2, 31, 35, 46, 100	rlimsqzlem 15682	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝_𝑟 1)
102	101	mptru 1544	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝_𝑟 1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 ℕ₀cn0 12524 ℝ⁺crp 13032 ⌊cfl 13827 !cfa 14309 abscabs 15270 ⇝_𝑟 crli 15518 logclog 26611 ↑_𝑐ccxp 26612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614

This theorem is referenced by: vmadivsum 27541

W3C validator