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Mirrors > Home > MPE Home > Th. List > chtleppi | Structured version Visualization version GIF version |
Description: Upper bound on the ΞΈ function. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
chtleppi | β’ (π΄ β β+ β (ΞΈβπ΄) β€ ((Οβπ΄) Β· (logβπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 13024 | . . . 4 β’ (π΄ β β+ β π΄ β β) | |
2 | ppifi 27066 | . . . 4 β’ (π΄ β β β ((0[,]π΄) β© β) β Fin) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π΄ β β+ β ((0[,]π΄) β© β) β Fin) |
4 | simpr 483 | . . . . . . 7 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β π β ((0[,]π΄) β© β)) | |
5 | 4 | elin2d 4201 | . . . . . 6 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β π β β) |
6 | prmnn 16654 | . . . . . 6 β’ (π β β β π β β) | |
7 | 5, 6 | syl 17 | . . . . 5 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β π β β) |
8 | 7 | nnrpd 13056 | . . . 4 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β π β β+) |
9 | 8 | relogcld 26585 | . . 3 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β (logβπ) β β) |
10 | relogcl 26537 | . . . 4 β’ (π΄ β β+ β (logβπ΄) β β) | |
11 | 10 | adantr 479 | . . 3 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β (logβπ΄) β β) |
12 | 4 | elin1d 4200 | . . . . . . 7 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β π β (0[,]π΄)) |
13 | 0re 11256 | . . . . . . . . 9 β’ 0 β β | |
14 | elicc2 13431 | . . . . . . . . 9 β’ ((0 β β β§ π΄ β β) β (π β (0[,]π΄) β (π β β β§ 0 β€ π β§ π β€ π΄))) | |
15 | 13, 1, 14 | sylancr 585 | . . . . . . . 8 β’ (π΄ β β+ β (π β (0[,]π΄) β (π β β β§ 0 β€ π β§ π β€ π΄))) |
16 | 15 | biimpa 475 | . . . . . . 7 β’ ((π΄ β β+ β§ π β (0[,]π΄)) β (π β β β§ 0 β€ π β§ π β€ π΄)) |
17 | 12, 16 | syldan 589 | . . . . . 6 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β (π β β β§ 0 β€ π β§ π β€ π΄)) |
18 | 17 | simp3d 1141 | . . . . 5 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β π β€ π΄) |
19 | 8 | reeflogd 26586 | . . . . 5 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β (expβ(logβπ)) = π) |
20 | reeflog 26542 | . . . . . 6 β’ (π΄ β β+ β (expβ(logβπ΄)) = π΄) | |
21 | 20 | adantr 479 | . . . . 5 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β (expβ(logβπ΄)) = π΄) |
22 | 18, 19, 21 | 3brtr4d 5184 | . . . 4 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β (expβ(logβπ)) β€ (expβ(logβπ΄))) |
23 | efle 16104 | . . . . 5 β’ (((logβπ) β β β§ (logβπ΄) β β) β ((logβπ) β€ (logβπ΄) β (expβ(logβπ)) β€ (expβ(logβπ΄)))) | |
24 | 9, 11, 23 | syl2anc 582 | . . . 4 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β ((logβπ) β€ (logβπ΄) β (expβ(logβπ)) β€ (expβ(logβπ΄)))) |
25 | 22, 24 | mpbird 256 | . . 3 β’ ((π΄ β β+ β§ π β ((0[,]π΄) β© β)) β (logβπ) β€ (logβπ΄)) |
26 | 3, 9, 11, 25 | fsumle 15787 | . 2 β’ (π΄ β β+ β Ξ£π β ((0[,]π΄) β© β)(logβπ) β€ Ξ£π β ((0[,]π΄) β© β)(logβπ΄)) |
27 | chtval 27070 | . . 3 β’ (π΄ β β β (ΞΈβπ΄) = Ξ£π β ((0[,]π΄) β© β)(logβπ)) | |
28 | 1, 27 | syl 17 | . 2 β’ (π΄ β β+ β (ΞΈβπ΄) = Ξ£π β ((0[,]π΄) β© β)(logβπ)) |
29 | ppival 27087 | . . . . 5 β’ (π΄ β β β (Οβπ΄) = (β―β((0[,]π΄) β© β))) | |
30 | 1, 29 | syl 17 | . . . 4 β’ (π΄ β β+ β (Οβπ΄) = (β―β((0[,]π΄) β© β))) |
31 | 30 | oveq1d 7441 | . . 3 β’ (π΄ β β+ β ((Οβπ΄) Β· (logβπ΄)) = ((β―β((0[,]π΄) β© β)) Β· (logβπ΄))) |
32 | 10 | recnd 11282 | . . . 4 β’ (π΄ β β+ β (logβπ΄) β β) |
33 | fsumconst 15778 | . . . 4 β’ ((((0[,]π΄) β© β) β Fin β§ (logβπ΄) β β) β Ξ£π β ((0[,]π΄) β© β)(logβπ΄) = ((β―β((0[,]π΄) β© β)) Β· (logβπ΄))) | |
34 | 3, 32, 33 | syl2anc 582 | . . 3 β’ (π΄ β β+ β Ξ£π β ((0[,]π΄) β© β)(logβπ΄) = ((β―β((0[,]π΄) β© β)) Β· (logβπ΄))) |
35 | 31, 34 | eqtr4d 2771 | . 2 β’ (π΄ β β+ β ((Οβπ΄) Β· (logβπ΄)) = Ξ£π β ((0[,]π΄) β© β)(logβπ΄)) |
36 | 26, 28, 35 | 3brtr4d 5184 | 1 β’ (π΄ β β+ β (ΞΈβπ΄) β€ ((Οβπ΄) Β· (logβπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β© cin 3948 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Fincfn 8972 βcc 11146 βcr 11147 0cc0 11148 Β· cmul 11153 β€ cle 11289 βcn 12252 β+crp 13016 [,]cicc 13369 β―chash 14331 Ξ£csu 15674 expce 16047 βcprime 16651 logclog 26516 ΞΈccht 27051 Οcppi 27054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-fi 9444 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ioo 13370 df-ioc 13371 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069 df-fac 14275 df-bc 14304 df-hash 14332 df-shft 15056 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-ef 16053 df-sin 16055 df-cos 16056 df-pi 16058 df-dvds 16241 df-prm 16652 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-rest 17413 df-topn 17414 df-0g 17432 df-gsum 17433 df-topgen 17434 df-pt 17435 df-prds 17438 df-xrs 17493 df-qtop 17498 df-imas 17499 df-xps 17501 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-mulg 19038 df-cntz 19282 df-cmn 19751 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22824 df-topon 22841 df-topsp 22863 df-bases 22877 df-cld 22951 df-ntr 22952 df-cls 22953 df-nei 23030 df-lp 23068 df-perf 23069 df-cn 23159 df-cnp 23160 df-haus 23247 df-tx 23494 df-hmeo 23687 df-fil 23778 df-fm 23870 df-flim 23871 df-flf 23872 df-xms 24254 df-ms 24255 df-tms 24256 df-cncf 24826 df-limc 25823 df-dv 25824 df-log 26518 df-cht 27057 df-ppi 27060 |
This theorem is referenced by: chtppilim 27436 |
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