Theorem chebbnd1 23380

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 10596	. . . . 5 |- 2 e. RR
2		pnfxr 11312	. . . . 5 |- +oo e. RR*
3		icossre 11596	. . . . 5 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( 2 [,) +oo ) C_ RR )
4	1, 2, 3	mp2an 672	. . . 4 |- ( 2 [,) +oo ) C_ RR
5	4	a1i 11	. . 3 |- ( T. -> ( 2 [,) +oo ) C_ RR )
6		elicopnf 11611	. . . . . . . . . 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 460	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
9		0red 9588	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 0 e. RR )
10		1re 9586	. . . . . . . . . 10 |- 1 e. RR
11	10	a1i 11	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 e. RR )
12		0lt1 10066	. . . . . . . . . 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 10693	. . . . . . . . . . 11 |- 1 < 2
16	15	a1i 11	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 1 < 2 )
17	7	simprbi 464	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
18	11, 14, 8, 16, 17	ltletrd 9732	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 < x )
19	9, 11, 8, 13, 18	lttrd 9733	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> 0 < x )
20	8, 19	elrpd 11245	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
21	8, 18	rplogcld 22737	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
22	20, 21	rpdivcld 11264	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
23		ppinncl 23171	. . . . . . . 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 11246	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
26	22, 25	rpdivcld 11264	. . . . 5 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
27	26	rpcnd 11249	. . . 4 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. CC )
28	27	adantl 466	. . 3 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. CC )
29		8re 10611	. . . 4 |- 8 e. RR
30	29	a1i 11	. . 3 |- ( T. -> 8 e. RR )
31		2rp 11216	. . . . . . . 8 |- 2 e. RR+
32		relogcl 22686	. . . . . . . 8 |- ( 2 e. RR+ -> ( log ` 2 ) e. RR )
33	31, 32	ax-mp 5	. . . . . . 7 |- ( log ` 2 ) e. RR
34		ere 13677	. . . . . . . . 9 |- _e e. RR
35	1, 34	remulcli 9601	. . . . . . . 8 |- ( 2 x. _e ) e. RR
36		2pos 10618	. . . . . . . . . 10 |- 0 < 2
37		epos 13792	. . . . . . . . . 10 |- 0 < _e
38	1, 34, 36, 37	mulgt0ii 9708	. . . . . . . . 9 |- 0 < ( 2 x. _e )
39	35, 38	gt0ne0ii 10080	. . . . . . . 8 |- ( 2 x. _e ) =/= 0
40	35, 39	rereccli 10300	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) e. RR
41	33, 40	resubcli 9872	. . . . . 6 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR
42		2t1e2 10675	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
43		egt2lt3 13791	. . . . . . . . . . . . 13 |- ( 2 < _e /\ _e < 3 )
44	43	simpli 458	. . . . . . . . . . . 12 |- 2 < _e
45	10, 1, 34	lttri 9701	. . . . . . . . . . . 12 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
46	15, 44, 45	mp2an 672	. . . . . . . . . . 11 |- 1 < _e
47	10, 34, 1	ltmul2i 10458	. . . . . . . . . . . 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 4463	. . . . . . . . 9 |- 2 < ( 2 x. _e )
51	1, 35, 36, 38	ltrecii 10453	. . . . . . . . 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 462	. . . . . . . . . . . 12 |- _e < 3
54		3lt4 10696	. . . . . . . . . . . 12 |- 3 < 4
55		3re 10600	. . . . . . . . . . . . 13 |- 3 e. RR
56		4re 10603	. . . . . . . . . . . . 13 |- 4 e. RR
57	34, 55, 56	lttri 9701	. . . . . . . . . . . 12 |- ( ( _e < 3 /\ 3 < 4 ) -> _e < 4 )
58	53, 54, 57	mp2an 672	. . . . . . . . . . 11 |- _e < 4
59		epr 13793	. . . . . . . . . . . 12 |- _e e. RR+
60		4pos 10622	. . . . . . . . . . . . 13 |- 0 < 4
61	56, 60	elrpii 11214	. . . . . . . . . . . 12 |- 4 e. RR+
62		logltb 22707	. . . . . . . . . . . 12 |- ( ( _e e. RR+ /\ 4 e. RR+ ) -> ( _e < 4 <-> ( log ` _e ) < ( log ` 4 ) ) )
63	59, 61, 62	mp2an 672	. . . . . . . . . . 11 |- ( _e < 4 <-> ( log ` _e ) < ( log ` 4 ) )
64	58, 63	mpbi 208	. . . . . . . . . 10 |- ( log ` _e ) < ( log ` 4 )
65		loge 22694	. . . . . . . . . 10 |- ( log ` _e ) = 1
66		sq2 12221	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
67	66	fveq2i 5862	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( log ` 4 )
68		2z 10887	. . . . . . . . . . . 12 |- 2 e. ZZ
69		relogexp 22703	. . . . . . . . . . . 12 |- ( ( 2 e. RR+ /\ 2 e. ZZ ) -> ( log ` ( 2 ^ 2 ) ) = ( 2 x. ( log ` 2 ) ) )
70	31, 68, 69	mp2an 672	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( 2 x. ( log ` 2 ) )
71	67, 70	eqtr3i 2493	. . . . . . . . . 10 |- ( log ` 4 ) = ( 2 x. ( log ` 2 ) )
72	64, 65, 71	3brtr3i 4469	. . . . . . . . 9 |- 1 < ( 2 x. ( log ` 2 ) )
73	1, 36	pm3.2i 455	. . . . . . . . . 10 |- ( 2 e. RR /\ 0 < 2 )
74		ltdivmul 10408	. . . . . . . . . 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 1319	. . . . . . . . 9 |- ( ( 1 / 2 ) < ( log ` 2 ) <-> 1 < ( 2 x. ( log ` 2 ) ) )
76	72, 75	mpbir 209	. . . . . . . 8 |- ( 1 / 2 ) < ( log ` 2 )
77		halfre 10745	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	40, 77, 33	lttri 9701	. . . . . . . 8 |- ( ( ( 1 / ( 2 x. _e ) ) < ( 1 / 2 ) /\ ( 1 / 2 ) < ( log ` 2 ) ) -> ( 1 / ( 2 x. _e ) ) < ( log ` 2 ) )
79	52, 76, 78	mp2an 672	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) < ( log ` 2 )
80	40, 33	posdifi 10094	. . . . . . 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 11214	. . . . 5 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR+
83		rerpdivcl 11238	. . . . 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 672	. . . 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 11217	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
87		rpge0 11223	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> 0 <_ ( ( x / ( log ` x ) ) / (π ` x ) ) )
88	86, 87	absidd 13205	. . . . . . 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 465	. . . . 5 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
91		eqid 2462	. . . . . . . . . 10 |- ( |_ ` ( x / 2 ) ) = ( |_ ` ( x / 2 ) )
92	91	chebbnd1lem3 23379	. . . . . . . . 9 |- ( ( x e. RR /\ 8 <_ x ) -> ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) < ( (π ` x ) x. ( ( log ` x ) / x ) ) )
93	8, 92	sylan 471	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) < ( (π ` x ) x. ( ( log ` x ) / x ) ) )
94	1	recni 9599	. . . . . . . . . 10 |- 2 e. CC
95		2ne0 10619	. . . . . . . . . 10 |- 2 =/= 0
96	41	recni 9599	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. CC
97	41, 81	gt0ne0ii 10080	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) =/= 0
98		recdiv 10241	. . . . . . . . . 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 673	. . . . . . . . 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 11249	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. CC )
102	24	nncnd 10543	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
103	22	rpne0d 11252	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) =/= 0 )
104	24	nnne0d 10571	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) =/= 0 )
105	101, 102, 103, 104	recdivd 10328	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
106	102, 101, 103	divrecd 10314	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) )
107	20	rpcnne0d 11256	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( x e. CC /\ x =/= 0 ) )
108	21	rpcnne0d 11256	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) )
109		recdiv 10241	. . . . . . . . . . . 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 661	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( x / ( log ` x ) ) ) = ( ( log ` x ) / x ) )
111	110	oveq2d 6293	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
112	105, 106, 111	3eqtrd 2507	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
113	112	adantr 465	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
114	93, 100, 113	3brtr4d 4472	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) < ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) )
115	26	adantr 465	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
116		elrp 11213	. . . . . . . . 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 10454	. . . . . . . . . 10 |- 0 < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) )
118		ltrec 10417	. . . . . . . . . 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 685	. . . . . . . . 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 11247	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
124		ltle 9664	. . . . . . 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 662	. . . . . 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 4462	. . . 4 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
128	127	adantl 466	. . 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 13310	. 2 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1) )
130	129	trud 1383	1 |- ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1)

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   T. wtru 1375   e. wcel 1762   =/= wne 2657   C_ wss 3471   class class class wbr 4442   |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484   x. cmul 9488   +oocpnf 9616   RR*cxr 9618   < clt 9619   <_ cle 9620   - cmin 9796   / cdiv 10197   NNcn 10527   2c2 10576   3c3 10577   4c4 10578   8c8 10582   ZZcz 10855   RR+crp 11211   [,)cico 11522   |_cfl 11886   ^cexp 12124   abscabs 13019   O(1)co1 13260   _eceu 13651   logclog 22665  πcppi 23090

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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-o1 13264  df-lo1 13265  df-sum 13460  df-ef 13656  df-e 13657  df-sin 13658  df-cos 13659  df-pi 13661  df-dvds 13839  df-gcd 13995  df-prm 14068  df-pc 14211  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001  df-log 22667  df-ppi 23096

This theorem is referenced by:  chtppilimlem2  23382  chto1lb  23386