Theorem chebbnd2 25166

Description: The Chebyshev bound, part 2: The function π ( x ) is eventually upper bounded by a positive constant times x / log ( x ). Alternatively stated, the function π ( x ) / ( x / log ( x ) ) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)

Assertion

Ref	Expression
chebbnd2	|- ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) e. O(1)

Proof of Theorem chebbnd2

Step	Hyp	Ref	Expression
1		ovexd 6680	. . . . 5 |- ( T. -> ( 2 [,) +oo ) e. _V )
2		ovexd 6680	. . . . 5 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / x ) e. _V )
3		ovexd 6680	. . . . 5 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) e. _V )
4		eqidd 2623	. . . . 5 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) )
5		simpr 477	. . . . . . . . . 10 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> x e. ( 2 [,) +oo ) )
6		2re 11090	. . . . . . . . . . 11 |- 2 e. RR
7		elicopnf 12269	. . . . . . . . . . 11 |- ( 2 e. RR -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) ) )
8	6, 7	ax-mp 5	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) )
9	5, 8	sylib 208	. . . . . . . . 9 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( x e. RR /\ 2 <_ x ) )
10		chtrpcl 24901	. . . . . . . . 9 |- ( ( x e. RR /\ 2 <_ x ) -> ( theta ` x ) e. RR+ )
11	9, 10	syl 17	. . . . . . . 8 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( theta ` x ) e. RR+ )
12	11	rpcnne0d 11881	. . . . . . 7 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) e. CC /\ ( theta ` x ) =/= 0 ) )
13		ppinncl 24900	. . . . . . . . . . 11 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
14	9, 13	syl 17	. . . . . . . . . 10 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> (π ` x ) e. NN )
15	14	nnrpd 11870	. . . . . . . . 9 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> (π ` x ) e. RR+ )
16	9	simpld 475	. . . . . . . . . 10 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> x e. RR )
17		1red 10055	. . . . . . . . . . 11 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 1 e. RR )
18	6	a1i 11	. . . . . . . . . . 11 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 2 e. RR )
19		1lt2 11194	. . . . . . . . . . . 12 |- 1 < 2
20	19	a1i 11	. . . . . . . . . . 11 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 1 < 2 )
21	9	simprd 479	. . . . . . . . . . 11 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 2 <_ x )
22	17, 18, 16, 20, 21	ltletrd 10197	. . . . . . . . . 10 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 1 < x )
23	16, 22	rplogcld 24375	. . . . . . . . 9 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( log ` x ) e. RR+ )
24	15, 23	rpmulcld 11888	. . . . . . . 8 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( (π ` x ) x. ( log ` x ) ) e. RR+ )
25	24	rpcnne0d 11881	. . . . . . 7 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( (π ` x ) x. ( log ` x ) ) e. CC /\ ( (π ` x ) x. ( log ` x ) ) =/= 0 ) )
26		recdiv 10731	. . . . . . 7 |- ( ( ( ( theta ` x ) e. CC /\ ( theta ` x ) =/= 0 ) /\ ( ( (π ` x ) x. ( log ` x ) ) e. CC /\ ( (π ` x ) x. ( log ` x ) ) =/= 0 ) ) -> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) )
27	12, 25, 26	syl2anc 693	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) )
28	27	mpteq2dva 4744	. . . . 5 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) ) )
29	1, 2, 3, 4, 28	offval2 6914	. . . 4 |- ( T. -> ( ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) oF x. ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) x. ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) ) ) )
30		0red 10041	. . . . . . . . . 10 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 0 e. RR )
31		2pos 11112	. . . . . . . . . . 11 |- 0 < 2
32	31	a1i 11	. . . . . . . . . 10 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 0 < 2 )
33	30, 18, 16, 32, 21	ltletrd 10197	. . . . . . . . 9 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 0 < x )
34	16, 33	elrpd 11869	. . . . . . . 8 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> x e. RR+ )
35	34	rpcnne0d 11881	. . . . . . 7 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( x e. CC /\ x =/= 0 ) )
36	24	rpcnd 11874	. . . . . . 7 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( (π ` x ) x. ( log ` x ) ) e. CC )
37		dmdcan 10735	. . . . . . 7 |- ( ( ( ( theta ` x ) e. CC /\ ( theta ` x ) =/= 0 ) /\ ( x e. CC /\ x =/= 0 ) /\ ( (π ` x ) x. ( log ` x ) ) e. CC ) -> ( ( ( theta ` x ) / x ) x. ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
38	12, 35, 36, 37	syl3anc 1326	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( ( theta ` x ) / x ) x. ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
39	15	rpcnd 11874	. . . . . . 7 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> (π ` x ) e. CC )
40	23	rpcnne0d 11881	. . . . . . 7 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) )
41		divdiv2 10737	. . . . . . 7 |- ( ( (π ` x ) e. CC /\ ( x e. CC /\ x =/= 0 ) /\ ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
42	39, 35, 40, 41	syl3anc 1326	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
43	38, 42	eqtr4d 2659	. . . . 5 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( ( theta ` x ) / x ) x. ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
44	43	mpteq2dva 4744	. . . 4 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) x. ( ( (π ` x ) x. ( log ` x ) ) / ( theta ` x ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
45	29, 44	eqtrd 2656	. . 3 |- ( T. -> ( ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) oF x. ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
46	34	ex 450	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) -> x e. RR+ ) )
47	46	ssrdv 3609	. . . . 5 |- ( T. -> ( 2 [,) +oo ) C_ RR+ )
48		chto1ub 25165	. . . . . 6 |- ( x e. RR+ |-> ( ( theta ` x ) / x ) ) e. O(1)
49	48	a1i 11	. . . . 5 |- ( T. -> ( x e. RR+ |-> ( ( theta ` x ) / x ) ) e. O(1) )
50	47, 49	o1res2 14294	. . . 4 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) e. O(1) )
51		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
52	51	a1i 11	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> 1 e. CC )
53	11, 24	rpdivcld 11889	. . . . . . 7 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
54	53	rpcnd 11874	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. CC )
55		pnfxr 10092	. . . . . . . . 9 |- +oo e. RR*
56		icossre 12254	. . . . . . . . 9 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( 2 [,) +oo ) C_ RR )
57	6, 55, 56	mp2an 708	. . . . . . . 8 |- ( 2 [,) +oo ) C_ RR
58		rlimconst 14275	. . . . . . . 8 |- ( ( ( 2 [,) +oo ) C_ RR /\ 1 e. CC ) -> ( x e. ( 2 [,) +oo ) |-> 1 ) ~~> r 1 )
59	57, 51, 58	mp2an 708	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) |-> 1 ) ~~> r 1
60	59	a1i 11	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> 1 ) ~~> r 1 )
61		chtppilim 25164	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1
62	61	a1i 11	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1 )
63		ax-1ne0 10005	. . . . . . 7 |- 1 =/= 0
64	63	a1i 11	. . . . . 6 |- ( T. -> 1 =/= 0 )
65	53	rpne0d 11877	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
66	52, 54, 60, 62, 64, 65	rlimdiv 14376	. . . . 5 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ~~> r ( 1 / 1 ) )
67		rlimo1 14347	. . . . 5 |- ( ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ~~> r ( 1 / 1 ) -> ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) e. O(1) )
68	66, 67	syl 17	. . . 4 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) e. O(1) )
69		o1mul 14345	. . . 4 |- ( ( ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) e. O(1) /\ ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) e. O(1) ) -> ( ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) oF x. ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ) e. O(1) )
70	50, 68, 69	syl2anc 693	. . 3 |- ( T. -> ( ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) oF x. ( x e. ( 2 [,) +oo ) |-> ( 1 / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ) e. O(1) )
71	45, 70	eqeltrrd 2702	. 2 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) e. O(1) )
72	71	trud 1493	1 |- ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) e. O(1)

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   2c2 11070   RR+crp 11832   [,)cico 12177   ~~> r crli 14216   O(1)co1 14217   logclog 24301   thetaccht 24817  πcppi 24820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-o1 14221  df-lo1 14222  df-sum 14417  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-cht 24823  df-ppi 24826

This theorem is referenced by: (None)