Theorem chebbnd1 22837

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 10492	. . . . 5 |- 2 e. RR
2		pnfxr 11193	. . . . 5 |- +oo e. RR*
3		icossre 11477	. . . . 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 11486	. . . . . . . . . 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 9488	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 0 e. RR )
10		1re 9486	. . . . . . . . . 10 |- 1 e. RR
11	10	a1i 11	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 e. RR )
12		0lt1 9963	. . . . . . . . . 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 10589	. . . . . . . . . . 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 9632	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 < x )
19	9, 11, 8, 13, 18	lttrd 9633	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> 0 < x )
20	8, 19	elrpd 11126	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
21	8, 18	rplogcld 22194	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
22	20, 21	rpdivcld 11145	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
23		ppinncl 22628	. . . . . . . 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 11127	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
26	22, 25	rpdivcld 11145	. . . . 5 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
27	26	rpcnd 11130	. . . 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 10507	. . . 4 |- 8 e. RR
30	29	a1i 11	. . 3 |- ( T. -> 8 e. RR )
31		2rp 11097	. . . . . . . 8 |- 2 e. RR+
32		relogcl 22143	. . . . . . . 8 |- ( 2 e. RR+ -> ( log ` 2 ) e. RR )
33	31, 32	ax-mp 5	. . . . . . 7 |- ( log ` 2 ) e. RR
34		ere 13476	. . . . . . . . 9 |- _e e. RR
35	1, 34	remulcli 9501	. . . . . . . 8 |- ( 2 x. _e ) e. RR
36		2pos 10514	. . . . . . . . . 10 |- 0 < 2
37		epos 13591	. . . . . . . . . 10 |- 0 < _e
38	1, 34, 36, 37	mulgt0ii 9608	. . . . . . . . 9 |- 0 < ( 2 x. _e )
39	35, 38	gt0ne0ii 9977	. . . . . . . 8 |- ( 2 x. _e ) =/= 0
40	35, 39	rereccli 10197	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) e. RR
41	33, 40	resubcli 9772	. . . . . 6 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR
42		2t1e2 10571	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
43		egt2lt3 13590	. . . . . . . . . . . . 13 |- ( 2 < _e /\ _e < 3 )
44	43	simpli 458	. . . . . . . . . . . 12 |- 2 < _e
45	10, 1, 34	lttri 9601	. . . . . . . . . . . 12 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
46	15, 44, 45	mp2an 672	. . . . . . . . . . 11 |- 1 < _e
47	10, 34, 1	ltmul2i 10355	. . . . . . . . . . . 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 4411	. . . . . . . . 9 |- 2 < ( 2 x. _e )
51	1, 35, 36, 38	ltrecii 10350	. . . . . . . . 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 10592	. . . . . . . . . . . 12 |- 3 < 4
55		3re 10496	. . . . . . . . . . . . 13 |- 3 e. RR
56		4re 10499	. . . . . . . . . . . . 13 |- 4 e. RR
57	34, 55, 56	lttri 9601	. . . . . . . . . . . 12 |- ( ( _e < 3 /\ 3 < 4 ) -> _e < 4 )
58	53, 54, 57	mp2an 672	. . . . . . . . . . 11 |- _e < 4
59		epr 13592	. . . . . . . . . . . 12 |- _e e. RR+
60		4pos 10518	. . . . . . . . . . . . 13 |- 0 < 4
61	56, 60	elrpii 11095	. . . . . . . . . . . 12 |- 4 e. RR+
62		logltb 22164	. . . . . . . . . . . 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 22151	. . . . . . . . . 10 |- ( log ` _e ) = 1
66		sq2 12063	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
67	66	fveq2i 5792	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( log ` 4 )
68		2z 10779	. . . . . . . . . . . 12 |- 2 e. ZZ
69		relogexp 22160	. . . . . . . . . . . 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 2482	. . . . . . . . . 10 |- ( log ` 4 ) = ( 2 x. ( log ` 2 ) )
72	64, 65, 71	3brtr3i 4417	. . . . . . . . 9 |- 1 < ( 2 x. ( log ` 2 ) )
73	1, 36	pm3.2i 455	. . . . . . . . . 10 |- ( 2 e. RR /\ 0 < 2 )
74		ltdivmul 10305	. . . . . . . . . 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 1315	. . . . . . . . 9 |- ( ( 1 / 2 ) < ( log ` 2 ) <-> 1 < ( 2 x. ( log ` 2 ) ) )
76	72, 75	mpbir 209	. . . . . . . 8 |- ( 1 / 2 ) < ( log ` 2 )
77		halfre 10641	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	40, 77, 33	lttri 9601	. . . . . . . 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 9991	. . . . . . 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 11095	. . . . 5 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR+
83		rerpdivcl 11119	. . . . 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 11098	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
87		rpge0 11104	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> 0 <_ ( ( x / ( log ` x ) ) / (π ` x ) ) )
88	86, 87	absidd 13011	. . . . . . 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 2451	. . . . . . . . . 10 |- ( |_ ` ( x / 2 ) ) = ( |_ ` ( x / 2 ) )
92	91	chebbnd1lem3 22836	. . . . . . . . 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 9499	. . . . . . . . . 10 |- 2 e. CC
95		2ne0 10515	. . . . . . . . . 10 |- 2 =/= 0
96	41	recni 9499	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. CC
97	41, 81	gt0ne0ii 9977	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) =/= 0
98		recdiv 10138	. . . . . . . . . 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 11130	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. CC )
102	24	nncnd 10439	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
103	22	rpne0d 11133	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) =/= 0 )
104	24	nnne0d 10467	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) =/= 0 )
105	101, 102, 103, 104	recdivd 10225	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
106	102, 101, 103	divrecd 10211	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) )
107	20	rpcnne0d 11137	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( x e. CC /\ x =/= 0 ) )
108	21	rpcnne0d 11137	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) )
109		recdiv 10138	. . . . . . . . . . . 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 6206	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
112	105, 106, 111	3eqtrd 2496	. . . . . . . . 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 4420	. . . . . . 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 11094	. . . . . . . . 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 10351	. . . . . . . . . 10 |- 0 < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) )
118		ltrec 10314	. . . . . . . . . 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 11128	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
124		ltle 9564	. . . . . . 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 4410	. . . 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 13116	. 2 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1) )
130	129	trud 1379	1 |- ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1)

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   T. wtru 1371   e. wcel 1758   =/= wne 2644   C_ wss 3426   class class class wbr 4390   |-> cmpt 4448   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384   x. cmul 9388   +oocpnf 9516   RR*cxr 9518   < clt 9519   <_ cle 9520   - cmin 9696   / cdiv 10094   NNcn 10423   2c2 10472   3c3 10473   4c4 10474   8c8 10478   ZZcz 10747   RR+crp 11092   [,)cico 11403   |_cfl 11741   ^cexp 11966   abscabs 12825   O(1)co1 13066   _eceu 13450   logclog 22122  πcppi 22547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-ioo 11405  df-ioc 11406  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-mod 11810  df-seq 11908  df-exp 11967  df-fac 12153  df-bc 12180  df-hash 12205  df-shft 12658  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-clim 13068  df-rlim 13069  df-o1 13070  df-lo1 13071  df-sum 13266  df-ef 13455  df-e 13456  df-sin 13457  df-cos 13458  df-pi 13460  df-dvds 13638  df-gcd 13793  df-prm 13866  df-pc 14006  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-fbas 17923  df-fg 17924  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cld 18739  df-ntr 18740  df-cls 18741  df-nei 18818  df-lp 18856  df-perf 18857  df-cn 18947  df-cnp 18948  df-haus 19035  df-tx 19251  df-hmeo 19444  df-fil 19535  df-fm 19627  df-flim 19628  df-flf 19629  df-xms 20011  df-ms 20012  df-tms 20013  df-cncf 20570  df-limc 21457  df-dv 21458  df-log 22124  df-ppi 22553

This theorem is referenced by:  chtppilimlem2  22839  chto1lb  22843