Theorem chebbnd1 21119

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 10025	. . . . 5 |- 2 e. RR
2		pnfxr 10669	. . . . 5 |- +oo e. RR*
3		icossre 10947	. . . . 5 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( 2 [,) +oo ) C_ RR )
4	1, 2, 3	mp2an 654	. . . 4 |- ( 2 [,) +oo ) C_ RR
5	4	a1i 11	. . 3 |- ( T. -> ( 2 [,) +oo ) C_ RR )
6		elicopnf 10956	. . . . . . . . . 10 |- ( 2 e. RR -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) ) )
7	1, 6	ax-mp 8	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) )
8	7	simplbi 447	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
9		0re 9047	. . . . . . . . . 10 |- 0 e. RR
10	9	a1i 11	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 0 e. RR )
11		1re 9046	. . . . . . . . . 10 |- 1 e. RR
12	11	a1i 11	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 e. RR )
13		0lt1 9506	. . . . . . . . . 10 |- 0 < 1
14	13	a1i 11	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 0 < 1 )
15	1	a1i 11	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 2 e. RR )
16		1lt2 10098	. . . . . . . . . . 11 |- 1 < 2
17	16	a1i 11	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 1 < 2 )
18	7	simprbi 451	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
19	12, 15, 8, 17, 18	ltletrd 9186	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 1 < x )
20	10, 12, 8, 14, 19	lttrd 9187	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> 0 < x )
21	8, 20	elrpd 10602	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
22	8, 19	rplogcld 20477	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
23	21, 22	rpdivcld 10621	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
24		ppinncl 20910	. . . . . . . 8 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
25	7, 24	sylbi 188	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. NN )
26	25	nnrpd 10603	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
27	23, 26	rpdivcld 10621	. . . . 5 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
28	27	rpcnd 10606	. . . 4 |- ( x e. ( 2 [,) +oo ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. CC )
29	28	adantl 453	. . 3 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. CC )
30		8re 10034	. . . 4 |- 8 e. RR
31	30	a1i 11	. . 3 |- ( T. -> 8 e. RR )
32		2rp 10573	. . . . . . . 8 |- 2 e. RR+
33		relogcl 20426	. . . . . . . 8 |- ( 2 e. RR+ -> ( log ` 2 ) e. RR )
34	32, 33	ax-mp 8	. . . . . . 7 |- ( log ` 2 ) e. RR
35		ere 12646	. . . . . . . . 9 |- _e e. RR
36	1, 35	remulcli 9060	. . . . . . . 8 |- ( 2 x. _e ) e. RR
37		2pos 10038	. . . . . . . . . 10 |- 0 < 2
38		epos 12761	. . . . . . . . . 10 |- 0 < _e
39	1, 35, 37, 38	mulgt0ii 9162	. . . . . . . . 9 |- 0 < ( 2 x. _e )
40	36, 39	gt0ne0ii 9519	. . . . . . . 8 |- ( 2 x. _e ) =/= 0
41	36, 40	rereccli 9735	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) e. RR
42	34, 41	resubcli 9319	. . . . . 6 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR
43	1	recni 9058	. . . . . . . . . . 11 |- 2 e. CC
44	43	mulid1i 9048	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
45		egt2lt3 12760	. . . . . . . . . . . . 13 |- ( 2 < _e /\ _e < 3 )
46	45	simpli 445	. . . . . . . . . . . 12 |- 2 < _e
47	11, 1, 35	lttri 9155	. . . . . . . . . . . 12 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
48	16, 46, 47	mp2an 654	. . . . . . . . . . 11 |- 1 < _e
49	11, 35, 1	ltmul2i 9888	. . . . . . . . . . . 12 |- ( 0 < 2 -> ( 1 < _e <-> ( 2 x. 1 ) < ( 2 x. _e ) ) )
50	37, 49	ax-mp 8	. . . . . . . . . . 11 |- ( 1 < _e <-> ( 2 x. 1 ) < ( 2 x. _e ) )
51	48, 50	mpbi 200	. . . . . . . . . 10 |- ( 2 x. 1 ) < ( 2 x. _e )
52	44, 51	eqbrtrri 4193	. . . . . . . . 9 |- 2 < ( 2 x. _e )
53	1, 36, 37, 39	ltrecii 9883	. . . . . . . . 9 |- ( 2 < ( 2 x. _e ) <-> ( 1 / ( 2 x. _e ) ) < ( 1 / 2 ) )
54	52, 53	mpbi 200	. . . . . . . 8 |- ( 1 / ( 2 x. _e ) ) < ( 1 / 2 )
55	45	simpri 449	. . . . . . . . . . . 12 |- _e < 3
56		3lt4 10101	. . . . . . . . . . . 12 |- 3 < 4
57		3re 10027	. . . . . . . . . . . . 13 |- 3 e. RR
58		4re 10029	. . . . . . . . . . . . 13 |- 4 e. RR
59	35, 57, 58	lttri 9155	. . . . . . . . . . . 12 |- ( ( _e < 3 /\ 3 < 4 ) -> _e < 4 )
60	55, 56, 59	mp2an 654	. . . . . . . . . . 11 |- _e < 4
61		epr 12762	. . . . . . . . . . . 12 |- _e e. RR+
62		4pos 10042	. . . . . . . . . . . . 13 |- 0 < 4
63	58, 62	elrpii 10571	. . . . . . . . . . . 12 |- 4 e. RR+
64		logltb 20447	. . . . . . . . . . . 12 |- ( ( _e e. RR+ /\ 4 e. RR+ ) -> ( _e < 4 <-> ( log ` _e ) < ( log ` 4 ) ) )
65	61, 63, 64	mp2an 654	. . . . . . . . . . 11 |- ( _e < 4 <-> ( log ` _e ) < ( log ` 4 ) )
66	60, 65	mpbi 200	. . . . . . . . . 10 |- ( log ` _e ) < ( log ` 4 )
67		loge 20434	. . . . . . . . . 10 |- ( log ` _e ) = 1
68		sq2 11432	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
69	68	fveq2i 5690	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( log ` 4 )
70		2z 10268	. . . . . . . . . . . 12 |- 2 e. ZZ
71		relogexp 20443	. . . . . . . . . . . 12 |- ( ( 2 e. RR+ /\ 2 e. ZZ ) -> ( log ` ( 2 ^ 2 ) ) = ( 2 x. ( log ` 2 ) ) )
72	32, 70, 71	mp2an 654	. . . . . . . . . . 11 |- ( log ` ( 2 ^ 2 ) ) = ( 2 x. ( log ` 2 ) )
73	69, 72	eqtr3i 2426	. . . . . . . . . 10 |- ( log ` 4 ) = ( 2 x. ( log ` 2 ) )
74	66, 67, 73	3brtr3i 4199	. . . . . . . . 9 |- 1 < ( 2 x. ( log ` 2 ) )
75	1, 37	pm3.2i 442	. . . . . . . . . 10 |- ( 2 e. RR /\ 0 < 2 )
76		ltdivmul 9838	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( log ` 2 ) e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( 1 / 2 ) < ( log ` 2 ) <-> 1 < ( 2 x. ( log ` 2 ) ) ) )
77	11, 34, 75, 76	mp3an 1279	. . . . . . . . 9 |- ( ( 1 / 2 ) < ( log ` 2 ) <-> 1 < ( 2 x. ( log ` 2 ) ) )
78	74, 77	mpbir 201	. . . . . . . 8 |- ( 1 / 2 ) < ( log ` 2 )
79	11	rehalfcli 10172	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
80	41, 79, 34	lttri 9155	. . . . . . . 8 |- ( ( ( 1 / ( 2 x. _e ) ) < ( 1 / 2 ) /\ ( 1 / 2 ) < ( log ` 2 ) ) -> ( 1 / ( 2 x. _e ) ) < ( log ` 2 ) )
81	54, 78, 80	mp2an 654	. . . . . . 7 |- ( 1 / ( 2 x. _e ) ) < ( log ` 2 )
82	41, 34	posdifi 9533	. . . . . . 7 |- ( ( 1 / ( 2 x. _e ) ) < ( log ` 2 ) <-> 0 < ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) )
83	81, 82	mpbi 200	. . . . . 6 |- 0 < ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) )
84	42, 83	elrpii 10571	. . . . 5 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR+
85		rerpdivcl 10595	. . . . 5 |- ( ( 2 e. RR /\ ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. RR+ ) -> ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) e. RR )
86	1, 84, 85	mp2an 654	. . . 4 |- ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) e. RR
87	86	a1i 11	. . 3 |- ( T. -> ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) e. RR )
88		rpre 10574	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
89		rpge0 10580	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> 0 <_ ( ( x / ( log ` x ) ) / (π ` x ) ) )
90	88, 89	absidd 12180	. . . . . . 7 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
91	27, 90	syl 16	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
92	91	adantr 452	. . . . 5 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( ( x / ( log ` x ) ) / (π ` x ) ) )
93		eqid 2404	. . . . . . . . . 10 |- ( |_ ` ( x / 2 ) ) = ( |_ ` ( x / 2 ) )
94	93	chebbnd1lem3 21118	. . . . . . . . 9 |- ( ( x e. RR /\ 8 <_ x ) -> ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) < ( (π ` x ) x. ( ( log ` x ) / x ) ) )
95	8, 94	sylan 458	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) < ( (π ` x ) x. ( ( log ` x ) / x ) ) )
96		2ne0 10039	. . . . . . . . . 10 |- 2 =/= 0
97	42	recni 9058	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) e. CC
98	42, 83	gt0ne0ii 9519	. . . . . . . . . 10 |- ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) =/= 0
99		recdiv 9676	. . . . . . . . . 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 ) )
100	43, 96, 97, 98, 99	mp4an 655	. . . . . . . . 9 |- ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) = ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 )
101	100	a1i 11	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) = ( ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) / 2 ) )
102	23	rpcnd 10606	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) e. CC )
103	25	nncnd 9972	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
104	23	rpne0d 10609	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( x / ( log ` x ) ) =/= 0 )
105	25	nnne0d 10000	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) =/= 0 )
106	102, 103, 104, 105	recdivd 9763	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
107	103, 102, 104	divrecd 9749	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) )
108	21	rpcnne0d 10613	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( x e. CC /\ x =/= 0 ) )
109	22	rpcnne0d 10613	. . . . . . . . . . . 12 |- ( x e. ( 2 [,) +oo ) -> ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) )
110		recdiv 9676	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) ) -> ( 1 / ( x / ( log ` x ) ) ) = ( ( log ` x ) / x ) )
111	108, 109, 110	syl2anc 643	. . . . . . . . . . 11 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( x / ( log ` x ) ) ) = ( ( log ` x ) / x ) )
112	111	oveq2d 6056	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( 1 / ( x / ( log ` x ) ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
113	106, 107, 112	3eqtrd 2440	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
114	113	adantr 452	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( (π ` x ) x. ( ( log ` x ) / x ) ) )
115	95, 101, 114	3brtr4d 4202	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( 1 / ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) ) < ( 1 / ( ( x / ( log ` x ) ) / (π ` x ) ) ) )
116	27	adantr 452	. . . . . . . 8 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
117		elrp 10570	. . . . . . . . 9 |- ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ <-> ( ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR /\ 0 < ( ( x / ( log ` x ) ) / (π ` x ) ) ) )
118	1, 42, 37, 83	divgt0ii 9884	. . . . . . . . . 10 |- 0 < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) )
119		ltrec 9847	. . . . . . . . . 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 ) ) ) ) )
120	86, 118, 119	mpanr12 667	. . . . . . . . 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 ) ) ) ) )
121	117, 120	sylbi 188	. . . . . . . 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 ) ) ) ) )
122	116, 121	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 ) ) ) ) )
123	115, 122	mpbird 224	. . . . . 6 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) < ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
124	116	rpred 10604	. . . . . . 7 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR )
125		ltle 9119	. . . . . . 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 ) ) ) ) ) )
126	124, 86, 125	sylancl 644	. . . . . 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 ) ) ) ) ) )
127	123, 126	mpd 15	. . . . 5 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
128	92, 127	eqbrtrd 4192	. . . 4 |- ( ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
129	128	adantl 453	. . 3 |- ( ( T. /\ ( x e. ( 2 [,) +oo ) /\ 8 <_ x ) ) -> ( abs ` ( ( x / ( log ` x ) ) / (π ` x ) ) ) <_ ( 2 / ( ( log ` 2 ) - ( 1 / ( 2 x. _e ) ) ) ) )
130	5, 29, 31, 87, 129	elo1d 12285	. 2 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O ( 1 ) )
131	130	trud 1329	1 |- ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O ( 1 )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   T. wtru 1322   = wceq 1649   e. wcel 1721   =/= wne 2567   C_ wss 3280   class class class wbr 4172   e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   x. cmul 8951   +oocpnf 9073   RR*cxr 9075   < clt 9076   <_ cle 9077   - cmin 9247   / cdiv 9633   NNcn 9956   2c2 10005   3c3 10006   4c4 10007   8c8 10011   ZZcz 10238   RR+crp 10568   [,)cico 10874   |_cfl 11156   ^cexp 11337   abscabs 11994   O ( 1 )co1 12235   _eceu 12620   logclog 20405  πcppi 20829

This theorem is referenced by:  chtppilimlem2  21121  chto1lb  21125

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-o1 12239  df-lo1 12240  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-ppi 20835