Theorem chebbnd1 24173

Description: The Chebyshev bound: The function π ( x ) is eventually lower bounded by a positive constant times x / log ( x ). Alternatively stated, the function ( x / log ( x ) ) / π ( x ) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)

Assertion

Ref	Expression
chebbnd1	|- ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1)

Proof of Theorem chebbnd1

Step	Hyp	Ref	Expression
1		2re 10679	. . . . 5 |- 2 e. RR
2		pnfxr 11412	. . . . 5 |- +oo e. RR*
3		icossre 11715	. . . . 5 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( 2 [,) +oo ) C_ RR )
4	1, 2, 3	mp2an 676	. . . 4 |- ( 2 [,) +oo ) C_ RR
5	4	a1i 11	. . 3 |- ( T. -> ( 2 [,) +oo ) C_ RR )
6		elicopnf 11730	. . . . . . . . . 10 |- ( 2 e. RR -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) ) )
7	1, 6	ax-mp 5	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) )
8	7	simplbi 461	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
9		0red 9643	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 0 e. RR )
10		1re 9641	. . . . . . . . . 10 |- 1 e. RR
11	10	a1i 11	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 e. RR )
12		0lt1 10135	. . . . . . . . . 10 |- 0 < 1
13	12	a1i 11	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 0 < 1 )
14	1	a1i 11	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 2 e. RR )
15		1lt2 10776	. . . . . . . . . . 11 |- 1 < 2
16	15	a1i 11	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 1 < 2 )
17	7	simprbi 465	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
18	11, 14, 8, 16, 17	ltletrd 9794	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 < x )
19	9, 11, 8, 13, 18	lttrd 9795	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> 0 < x )
20	8, 19	elrpd 11338	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
21	8, 18	rplogcld 23443	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
22	20, 21	rpdivcld 11358	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
23		ppinncl 23964	. . . . . . . 8 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
24	7, 23	sylbi 198	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. NN )
25	24	nnrpd 11339	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
26	22, 25	rpdivcld 11358	. . . . 5 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
27	26	rpcnd 11343	. . . 4 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. CC )
28	27	adantl 467	. . 3 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. CC )
29		8re 10694	. . . 4 |- 8 e. RR
30	29	a1i 11	. . 3 |- ( T. -> 8 e. RR )
31		2rp 11307	. . . . . . . 8 |- 2 e. RR+
32		relogcl 23390	. . . . . . . 8 |- ( 2 e. RR+ -> ( log ` 2 ) e. RR )
33	31, 32	ax-mp 5	. . . . . . 7 |- ( log ` 2 ) e. RR
34		ere 14121	. . . . . . . . 9 |- _e e. RR
35	1, 34	remulcli 9656	. . . . . . . 8 |- ( 2 x. _e ) e. RR
36		2pos 10701	. . . . . . . . . 10 |- 0 < 2
37		epos 14237	. . . . . . . . . 10 |- 0 < _e
38	1, 34, 36, 37	mulgt0ii 9767	. . . . . . . . 9 |- 0 < ( 2 x. _e )
39	35, 38	gt0ne0ii 10149	. . . . . . . 8 |- ( 2 x. _e ) =/= 0
40	35, 39	rereccli 10371	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) e. RR
41	33, 40	resubcli 9935	. . . . . 6 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR
42		2t1e2 10758	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
43		egt2lt3 14236	. . . . . . . . . . . . 13 |- ( 2 < _e /\ _e < 3 )
44	43	simpli 459	. . . . . . . . . . . 12 |- 2 < _e
45	10, 1, 34	lttri 9759	. . . . . . . . . . . 12 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
46	15, 44, 45	mp2an 676	. . . . . . . . . . 11 |- 1 < _e
47	10, 34, 1	ltmul2i 10528	. . . . . . . . . . . 12 |- ( 0 < 2 -> ( 1 < _e <-> ( 2 x. 1 ) < ( 2 x. _e ) ) )
48	36, 47	ax-mp 5	. . . . . . . . . . 11 |- ( 1 < _e <-> ( 2 x. 1 ) < ( 2 x. _e ) )
49	46, 48	mpbi 211	. . . . . . . . . 10 |- ( 2 x. 1 ) < ( 2 x. _e )
50	42, 49	eqbrtrri 4447	. . . . . . . . 9 |- 2 < ( 2 x. _e )
51	1, 35, 36, 38	ltrecii 10523	. . . . . . . . 9 |- ( 2 < ( 2 x. _e ) <-> ( 1 / ( 2 x. _e ) ) < ( 1 / 2 ) )
52	50, 51	mpbi 211	. . . . . . . 8 |- ( 1 / ( 2 x. _e ) ) < ( 1 / 2 )
53	43	simpri 463	. . . . . . . . . . . 12 |- _e < 3
54		3lt4 10779	. . . . . . . . . . . 12 |- 3 < 4
55		3re 10683	. . . . . . . . . . . . 13 |- 3 e. RR
56		4re 10686	. . . . . . . . . . . . 13 |- 4 e. RR
57	34, 55, 56	lttri 9759	. . . . . . . . . . . 12 |- ( ( _e < 3 /\ 3 < 4 ) -> _e < 4 )
58	53, 54, 57	mp2an 676	. . . . . . . . . . 11 |- _e < 4
59		epr 14238	. . . . . . . . . . . 12 |- _e e. RR+
60		4pos 10705	. . . . . . . . . . . . 13 |- 0 < 4
61	56, 60	elrpii 11305	. . . . . . . . . . . 12 |- 4 e. RR+
62		logltb 23414	. . . . . . . . . . . 12 |- ( ( _e e. RR+ /\ 4 e. RR+ ) -> ( _e < 4 <-> ( log ` _e ) < ( log ` 4 ) ) )
63	59, 61, 62	mp2an 676	. . . . . . . . . . 11 |- ( _e < 4 <-> ( log ` _e ) < ( log ` 4 ) )
64	58, 63	mpbi 211	. . . . . . . . . 10 |- ( log ` _e ) < ( log ` 4 )
65		loge 23401	. . . . . . . . . 10 |- ( log ` _e ) = 1
66		sq2 12368	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
67	66	fveq2i 5884	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( log ` 4 )
68		2z 10969	. . . . . . . . . . . 12 |- 2 e. ZZ
69		relogexp 23410	. . . . . . . . . . . 12 |- ( ( 2 e. RR+ /\ 2 e. ZZ ) -> ( log ` ( 2 ^ 2 ) ) = ( 2 x. ( log ` 2 ) ) )
70	31, 68, 69	mp2an 676	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( 2 x. ( log ` 2 ) )
71	67, 70	eqtr3i 2460	. . . . . . . . . 10 |- ( log ` 4 ) = ( 2 x. ( log ` 2 ) )
72	64, 65, 71	3brtr3i 4453	. . . . . . . . 9 |- 1 < ( 2 x. ( log ` 2 ) )
73	1, 36	pm3.2i 456	. . . . . . . . . 10 |- ( 2 e. RR /\ 0 < 2 )
74		ltdivmul 10479	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( log ` 2 ) e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( 1 / 2 ) < ( log ` 2 ) <-> 1 < ( 2 x. ( log ` 2 ) ) ) )
75	10, 33, 73, 74	mp3an 1360	. . . . . . . . 9 |- ( ( 1 / 2 ) < ( log ` 2 ) <-> 1 < ( 2 x. ( log ` 2 ) ) )
76	72, 75	mpbir 212	. . . . . . . 8 |- ( 1 / 2 ) < ( log ` 2 )
77		halfre 10828	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	40, 77, 33	lttri 9759	. . . . . . . 8 |- ( ( ( 1 / ( 2 x. _e ) ) < ( 1 / 2 ) /\ ( 1 / 2 ) < ( log ` 2 ) ) -> ( 1 / ( 2 x. _e ) ) < ( log ` 2 ) )
79	52, 76, 78	mp2an 676	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) < ( log ` 2 )
80	40, 33	posdifi 10163	. . . . . . 7 |- ( ( 1 / ( 2 x. _e ) ) < ( log ` 2 ) <-> 0 < ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) )
81	79, 80	mpbi 211	. . . . . 6 |- 0 < ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) )
82	41, 81	elrpii 11305	. . . . 5 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR+
83		rerpdivcl 11330	. . . . 5 |- ( ( 2 e. RR /\ ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR+ ) -> ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) e. RR )
84	1, 82, 83	mp2an 676	. . . 4 |- ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) e. RR
85	84	a1i 11	. . 3 |- ( T. -> ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) e. RR )
86		rpre 11308	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
87		rpge0 11314	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> 0 <_ ( ( x / ( log ` x ) ) / (π ` x ) ) )
88	86, 87	absidd 13463	. . . . . . 7 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
89	26, 88	syl 17	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
90	89	adantr 466	. . . . 5 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
91		eqid 2429	. . . . . . . . . 10 |- ( |_ ` ( x / 2 ) ) = ( |_ ` ( x / 2 ) )
92	91	chebbnd1lem3 24172	. . . . . . . . 9 |- ( ( x e. RR /\ 8 <_ x ) -> ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) < ( (π ` x ) x. ( ( log ` x ) / x ) ) )
93	8, 92	sylan 473	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) < ( (π ` x ) x. ( ( log ` x ) / x ) ) )
94	1	recni 9654	. . . . . . . . . 10 |- 2 e. CC
95		2ne0 10702	. . . . . . . . . 10 |- 2 =/= 0
96	41	recni 9654	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. CC
97	41, 81	gt0ne0ii 10149	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) =/= 0
98		recdiv 10312	. . . . . . . . . 10 |- ( ( ( 2 e. CC /\ 2 =/= 0 ) /\ ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. CC /\ ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) =/= 0 ) ) -> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) = ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) )
99	94, 95, 96, 97, 98	mp4an 677	. . . . . . . . 9 |- ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) = ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 )
100	99	a1i 11	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) = ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) )
101	22	rpcnd 11343	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. CC )
102	24	nncnd 10625	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
103	22	rpne0d 11346	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) =/= 0 )
104	24	nnne0d 10654	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) =/= 0 )
105	101, 102, 103, 104	recdivd 10399	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
106	102, 101, 103	divrecd 10385	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) )
107	20	rpcnne0d 11350	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( x e. CC /\ x =/= 0 ) )
108	21	rpcnne0d 11350	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) )
109		recdiv 10312	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) ) -> ( 1 / ( x / ( log ` x ) ) ) = ( ( log ` x ) / x ) )
110	107, 108, 109	syl2anc 665	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( x / ( log ` x ) ) ) = ( ( log ` x ) / x ) )
111	110	oveq2d 6321	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
112	105, 106, 111	3eqtrd 2474	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
113	112	adantr 466	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
114	93, 100, 113	3brtr4d 4456	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) < ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) )
115	26	adantr 466	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
116		elrp 11304	. . . . . . . . 9 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ <-> ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR /\ 0 < ( ( x / ( log ` x ) ) / (π ` x ) ) ) )
117	1, 41, 36, 81	divgt0ii 10524	. . . . . . . . . 10 |- 0 < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) )
118		ltrec 10487	. . . . . . . . . 10 |- ( ( ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR /\ 0 < ( ( x / ( log ` x ) ) / (π ` x ) ) ) /\ ( ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) e. RR /\ 0 < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) ) -> ( ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) <-> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) < ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) ) )
119	84, 117, 118	mpanr12 689	. . . . . . . . 9 |- ( ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR /\ 0 < ( ( x / ( log ` x ) ) / (π ` x ) ) ) -> ( ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) <-> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) < ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) ) )
120	116, 119	sylbi 198	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) <-> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) < ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) ) )
121	115, 120	syl 17	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) <-> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) < ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) ) )
122	114, 121	mpbird 235	. . . . . 6 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
123	115	rpred 11341	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
124		ltle 9721	. . . . . . 7 |- ( ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR /\ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) e. RR ) -> ( ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) )
125	123, 84, 124	sylancl 666	. . . . . 6 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) )
126	122, 125	mpd 15	. . . . 5 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
127	90, 126	eqbrtrd 4446	. . . 4 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
128	127	adantl 467	. . 3 |- ( ( T. /\ ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
129	5, 28, 30, 85, 128	elo1d 13578	. 2 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1) )
130	129	trud 1446	1 |- ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1)

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   T. wtru 1438   e. wcel 1870   =/= wne 2625   C_ wss 3442   class class class wbr 4426   |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539   x. cmul 9543   +oocpnf 9671   RR*cxr 9673   < clt 9674   <_ cle 9675   - cmin 9859   / cdiv 10268   NNcn 10609   2c2 10659   3c3 10660   4c4 10661   8c8 10665   ZZcz 10937   RR+crp 11302   [,)cico 11637   |_cfl 12023   ^cexp 12269   abscabs 13276   O(1)co1 13528   _eceu 14093   logclog 23369  πcppi 23883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-o1 13532  df-lo1 13533  df-sum 13731  df-ef 14099  df-e 14100  df-sin 14101  df-cos 14102  df-pi 14104  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699  df-log 23371  df-ppi 23889

This theorem is referenced by:  chtppilimlem2  24175  chto1lb  24179