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Mirrors > Home > MPE Home > Th. List > chtleppi | Structured version Visualization version GIF version |
Description: Upper bound on the θ function. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
chtleppi | ⊢ (𝐴 ∈ ℝ+ → (θ‘𝐴) ≤ ((π‘𝐴) · (log‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 12391 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | ppifi 25677 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((0[,]𝐴) ∩ ℙ) ∈ Fin) |
4 | simpr 487 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) | |
5 | 4 | elin2d 4176 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ) |
6 | prmnn 16012 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ) |
8 | 7 | nnrpd 12423 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+) |
9 | 8 | relogcld 25200 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ) |
10 | relogcl 25153 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ) |
12 | 4 | elin1d 4175 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴)) |
13 | 0re 10637 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
14 | elicc2 12795 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))) | |
15 | 13, 1, 14 | sylancr 589 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))) |
16 | 15 | biimpa 479 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)) |
17 | 12, 16 | syldan 593 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)) |
18 | 17 | simp3d 1140 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴) |
19 | 8 | reeflogd 25201 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝) |
20 | reeflog 25158 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴) | |
21 | 20 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝐴)) = 𝐴) |
22 | 18, 19, 21 | 3brtr4d 5091 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴))) |
23 | efle 15465 | . . . . 5 ⊢ (((log‘𝑝) ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → ((log‘𝑝) ≤ (log‘𝐴) ↔ (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴)))) | |
24 | 9, 11, 23 | syl2anc 586 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) ≤ (log‘𝐴) ↔ (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴)))) |
25 | 22, 24 | mpbird 259 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ≤ (log‘𝐴)) |
26 | 3, 9, 11, 25 | fsumle 15148 | . 2 ⊢ (𝐴 ∈ ℝ+ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴)) |
27 | chtval 25681 | . . 3 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) | |
28 | 1, 27 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
29 | ppival 25698 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ))) | |
30 | 1, 29 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ))) |
31 | 30 | oveq1d 7165 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((π‘𝐴) · (log‘𝐴)) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴))) |
32 | 10 | recnd 10663 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℂ) |
33 | fsumconst 15139 | . . . 4 ⊢ ((((0[,]𝐴) ∩ ℙ) ∈ Fin ∧ (log‘𝐴) ∈ ℂ) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴))) | |
34 | 3, 32, 33 | syl2anc 586 | . . 3 ⊢ (𝐴 ∈ ℝ+ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴))) |
35 | 31, 34 | eqtr4d 2859 | . 2 ⊢ (𝐴 ∈ ℝ+ → ((π‘𝐴) · (log‘𝐴)) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴)) |
36 | 26, 28, 35 | 3brtr4d 5091 | 1 ⊢ (𝐴 ∈ ℝ+ → (θ‘𝐴) ≤ ((π‘𝐴) · (log‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∩ cin 3935 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 ℂcc 10529 ℝcr 10530 0cc0 10531 · cmul 10536 ≤ cle 10670 ℕcn 11632 ℝ+crp 12383 [,]cicc 12735 ♯chash 13684 Σcsu 15036 expce 15409 ℙcprime 16009 logclog 25132 θccht 25662 πcppi 25665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-dvds 15602 df-prm 16010 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-cht 25668 df-ppi 25671 |
This theorem is referenced by: chtppilim 26045 |
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