Theorem chebbnd1 22680

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 10387	. . . . 5 |- 2 e. RR
2		pnfxr 11088	. . . . 5 |- +oo e. RR*
3		icossre 11372	. . . . 5 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( 2 [,) +oo ) C_ RR )
4	1, 2, 3	mp2an 667	. . . 4 |- ( 2 [,) +oo ) C_ RR
5	4	a1i 11	. . 3 |- ( T. -> ( 2 [,) +oo ) C_ RR )
6		elicopnf 11381	. . . . . . . . . 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 457	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
9		0red 9383	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 0 e. RR )
10		1re 9381	. . . . . . . . . 10 |- 1 e. RR
11	10	a1i 11	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 e. RR )
12		0lt1 9858	. . . . . . . . . 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 10484	. . . . . . . . . . 11 |- 1 < 2
16	15	a1i 11	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 1 < 2 )
17	7	simprbi 461	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
18	11, 14, 8, 16, 17	ltletrd 9527	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 < x )
19	9, 11, 8, 13, 18	lttrd 9528	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> 0 < x )
20	8, 19	elrpd 11021	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
21	8, 18	rplogcld 22037	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
22	20, 21	rpdivcld 11040	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
23		ppinncl 22471	. . . . . . . 8 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
24	7, 23	sylbi 195	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. NN )
25	24	nnrpd 11022	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
26	22, 25	rpdivcld 11040	. . . . 5 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
27	26	rpcnd 11025	. . . 4 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. CC )
28	27	adantl 463	. . 3 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. CC )
29		8re 10402	. . . 4 |- 8 e. RR
30	29	a1i 11	. . 3 |- ( T. -> 8 e. RR )
31		2rp 10992	. . . . . . . 8 |- 2 e. RR+
32		relogcl 21986	. . . . . . . 8 |- ( 2 e. RR+ -> ( log ` 2 ) e. RR )
33	31, 32	ax-mp 5	. . . . . . 7 |- ( log ` 2 ) e. RR
34		ere 13370	. . . . . . . . 9 |- _e e. RR
35	1, 34	remulcli 9396	. . . . . . . 8 |- ( 2 x. _e ) e. RR
36		2pos 10409	. . . . . . . . . 10 |- 0 < 2
37		epos 13485	. . . . . . . . . 10 |- 0 < _e
38	1, 34, 36, 37	mulgt0ii 9503	. . . . . . . . 9 |- 0 < ( 2 x. _e )
39	35, 38	gt0ne0ii 9872	. . . . . . . 8 |- ( 2 x. _e ) =/= 0
40	35, 39	rereccli 10092	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) e. RR
41	33, 40	resubcli 9667	. . . . . 6 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR
42		2t1e2 10466	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
43		egt2lt3 13484	. . . . . . . . . . . . 13 |- ( 2 < _e /\ _e < 3 )
44	43	simpli 455	. . . . . . . . . . . 12 |- 2 < _e
45	10, 1, 34	lttri 9496	. . . . . . . . . . . 12 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
46	15, 44, 45	mp2an 667	. . . . . . . . . . 11 |- 1 < _e
47	10, 34, 1	ltmul2i 10250	. . . . . . . . . . . 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 208	. . . . . . . . . 10 |- ( 2 x. 1 ) < ( 2 x. _e )
50	42, 49	eqbrtrri 4310	. . . . . . . . 9 |- 2 < ( 2 x. _e )
51	1, 35, 36, 38	ltrecii 10245	. . . . . . . . 9 |- ( 2 < ( 2 x. _e ) <-> ( 1 / ( 2 x. _e ) ) < ( 1 / 2 ) )
52	50, 51	mpbi 208	. . . . . . . 8 |- ( 1 / ( 2 x. _e ) ) < ( 1 / 2 )
53	43	simpri 459	. . . . . . . . . . . 12 |- _e < 3
54		3lt4 10487	. . . . . . . . . . . 12 |- 3 < 4
55		3re 10391	. . . . . . . . . . . . 13 |- 3 e. RR
56		4re 10394	. . . . . . . . . . . . 13 |- 4 e. RR
57	34, 55, 56	lttri 9496	. . . . . . . . . . . 12 |- ( ( _e < 3 /\ 3 < 4 ) -> _e < 4 )
58	53, 54, 57	mp2an 667	. . . . . . . . . . 11 |- _e < 4
59		epr 13486	. . . . . . . . . . . 12 |- _e e. RR+
60		4pos 10413	. . . . . . . . . . . . 13 |- 0 < 4
61	56, 60	elrpii 10990	. . . . . . . . . . . 12 |- 4 e. RR+
62		logltb 22007	. . . . . . . . . . . 12 |- ( ( _e e. RR+ /\ 4 e. RR+ ) -> ( _e < 4 <-> ( log ` _e ) < ( log ` 4 ) ) )
63	59, 61, 62	mp2an 667	. . . . . . . . . . 11 |- ( _e < 4 <-> ( log ` _e ) < ( log ` 4 ) )
64	58, 63	mpbi 208	. . . . . . . . . 10 |- ( log ` _e ) < ( log ` 4 )
65		loge 21994	. . . . . . . . . 10 |- ( log ` _e ) = 1
66		sq2 11958	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
67	66	fveq2i 5691	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( log ` 4 )
68		2z 10674	. . . . . . . . . . . 12 |- 2 e. ZZ
69		relogexp 22003	. . . . . . . . . . . 12 |- ( ( 2 e. RR+ /\ 2 e. ZZ ) -> ( log ` ( 2 ^ 2 ) ) = ( 2 x. ( log ` 2 ) ) )
70	31, 68, 69	mp2an 667	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( 2 x. ( log ` 2 ) )
71	67, 70	eqtr3i 2463	. . . . . . . . . 10 |- ( log ` 4 ) = ( 2 x. ( log ` 2 ) )
72	64, 65, 71	3brtr3i 4316	. . . . . . . . 9 |- 1 < ( 2 x. ( log ` 2 ) )
73	1, 36	pm3.2i 452	. . . . . . . . . 10 |- ( 2 e. RR /\ 0 < 2 )
74		ltdivmul 10200	. . . . . . . . . 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 1309	. . . . . . . . 9 |- ( ( 1 / 2 ) < ( log ` 2 ) <-> 1 < ( 2 x. ( log ` 2 ) ) )
76	72, 75	mpbir 209	. . . . . . . 8 |- ( 1 / 2 ) < ( log ` 2 )
77		halfre 10536	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	40, 77, 33	lttri 9496	. . . . . . . 8 |- ( ( ( 1 / ( 2 x. _e ) ) < ( 1 / 2 ) /\ ( 1 / 2 ) < ( log ` 2 ) ) -> ( 1 / ( 2 x. _e ) ) < ( log ` 2 ) )
79	52, 76, 78	mp2an 667	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) < ( log ` 2 )
80	40, 33	posdifi 9886	. . . . . . 7 |- ( ( 1 / ( 2 x. _e ) ) < ( log ` 2 ) <-> 0 < ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) )
81	79, 80	mpbi 208	. . . . . 6 |- 0 < ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) )
82	41, 81	elrpii 10990	. . . . 5 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR+
83		rerpdivcl 11014	. . . . 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 667	. . . 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 10993	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
87		rpge0 10999	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> 0 <_ ( ( x / ( log ` x ) ) / (π ` x ) ) )
88	86, 87	absidd 12905	. . . . . . 7 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
89	26, 88	syl 16	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
90	89	adantr 462	. . . . 5 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
91		eqid 2441	. . . . . . . . . 10 |- ( |_ ` ( x / 2 ) ) = ( |_ ` ( x / 2 ) )
92	91	chebbnd1lem3 22679	. . . . . . . . 9 |- ( ( x e. RR /\ 8 <_ x ) -> ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) < ( (π ` x ) x. ( ( log ` x ) / x ) ) )
93	8, 92	sylan 468	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) < ( (π ` x ) x. ( ( log ` x ) / x ) ) )
94	1	recni 9394	. . . . . . . . . 10 |- 2 e. CC
95		2ne0 10410	. . . . . . . . . 10 |- 2 =/= 0
96	41	recni 9394	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. CC
97	41, 81	gt0ne0ii 9872	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) =/= 0
98		recdiv 10033	. . . . . . . . . 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 668	. . . . . . . . 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 11025	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. CC )
102	24	nncnd 10334	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
103	22	rpne0d 11028	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) =/= 0 )
104	24	nnne0d 10362	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) =/= 0 )
105	101, 102, 103, 104	recdivd 10120	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
106	102, 101, 103	divrecd 10106	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) )
107	20	rpcnne0d 11032	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( x e. CC /\ x =/= 0 ) )
108	21	rpcnne0d 11032	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) )
109		recdiv 10033	. . . . . . . . . . . 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 656	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( x / ( log ` x ) ) ) = ( ( log ` x ) / x ) )
111	110	oveq2d 6106	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
112	105, 106, 111	3eqtrd 2477	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
113	112	adantr 462	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
114	93, 100, 113	3brtr4d 4319	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) < ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) )
115	26	adantr 462	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
116		elrp 10989	. . . . . . . . 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 10246	. . . . . . . . . 10 |- 0 < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) )
118		ltrec 10209	. . . . . . . . . 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 680	. . . . . . . . 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 195	. . . . . . . 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 16	. . . . . . 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 232	. . . . . 6 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
123	115	rpred 11023	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
124		ltle 9459	. . . . . . 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 657	. . . . . 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 4309	. . . 4 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
128	127	adantl 463	. . 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 13010	. 2 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1) )
130	129	trud 1373	1 |- ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1)

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1364   T. wtru 1365   e. wcel 1761   =/= wne 2604   C_ wss 3325   class class class wbr 4289   e. cmpt 4347   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   x. cmul 9283   +oocpnf 9411   RR*cxr 9413   < clt 9414   <_ cle 9415   - cmin 9591   / cdiv 9989   NNcn 10318   2c2 10367   3c3 10368   4c4 10369   8c8 10373   ZZcz 10642   RR+crp 10987   [,)cico 11298   |_cfl 11636   ^cexp 11861   abscabs 12719   O(1)co1 12960   _eceu 13344   logclog 21965  πcppi 22390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-o1 12964  df-lo1 12965  df-sum 13160  df-ef 13349  df-e 13350  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-ppi 22396

This theorem is referenced by:  chtppilimlem2  22682  chto1lb  22686