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Mirrors > Home > MPE Home > Th. List > vmalelog | Structured version Visualization version GIF version |
Description: The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.) |
Ref | Expression |
---|---|
vmalelog | ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) ≤ (log‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5062 | . 2 ⊢ ((Λ‘𝐴) = 0 → ((Λ‘𝐴) ≤ (log‘𝐴) ↔ 0 ≤ (log‘𝐴))) | |
2 | isppw2 25690 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝐴 = (𝑝↑𝑘))) | |
3 | prmnn 16013 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ) | |
4 | 3 | nnrpd 12423 | . . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ+) |
5 | 4 | adantr 483 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℝ+) |
6 | 5 | relogcld 25204 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℝ) |
7 | nnre 11638 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ) | |
8 | 7 | adantl 484 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ) |
9 | log1 25167 | . . . . . . . . 9 ⊢ (log‘1) = 0 | |
10 | 3 | adantr 483 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℕ) |
11 | 10 | nnge1d 11679 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑝) |
12 | 1rp 12387 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ+ | |
13 | logleb 25184 | . . . . . . . . . . 11 ⊢ ((1 ∈ ℝ+ ∧ 𝑝 ∈ ℝ+) → (1 ≤ 𝑝 ↔ (log‘1) ≤ (log‘𝑝))) | |
14 | 12, 5, 13 | sylancr 589 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (1 ≤ 𝑝 ↔ (log‘1) ≤ (log‘𝑝))) |
15 | 11, 14 | mpbid 234 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (log‘1) ≤ (log‘𝑝)) |
16 | 9, 15 | eqbrtrrid 5095 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 0 ≤ (log‘𝑝)) |
17 | nnge1 11659 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 1 ≤ 𝑘) | |
18 | 17 | adantl 484 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑘) |
19 | 6, 8, 16, 18 | lemulge12d 11571 | . . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ≤ (𝑘 · (log‘𝑝))) |
20 | vmappw 25691 | . . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝)) | |
21 | nnz 11998 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
22 | relogexp 25177 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝))) | |
23 | 4, 21, 22 | syl2an 597 | . . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝))) |
24 | 19, 20, 23 | 3brtr4d 5091 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝↑𝑘)) ≤ (log‘(𝑝↑𝑘))) |
25 | fveq2 6663 | . . . . . . 7 ⊢ (𝐴 = (𝑝↑𝑘) → (Λ‘𝐴) = (Λ‘(𝑝↑𝑘))) | |
26 | fveq2 6663 | . . . . . . 7 ⊢ (𝐴 = (𝑝↑𝑘) → (log‘𝐴) = (log‘(𝑝↑𝑘))) | |
27 | 25, 26 | breq12d 5072 | . . . . . 6 ⊢ (𝐴 = (𝑝↑𝑘) → ((Λ‘𝐴) ≤ (log‘𝐴) ↔ (Λ‘(𝑝↑𝑘)) ≤ (log‘(𝑝↑𝑘)))) |
28 | 24, 27 | syl5ibrcom 249 | . . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (𝐴 = (𝑝↑𝑘) → (Λ‘𝐴) ≤ (log‘𝐴))) |
29 | 28 | rexlimivv 3291 | . . . 4 ⊢ (∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝐴 = (𝑝↑𝑘) → (Λ‘𝐴) ≤ (log‘𝐴)) |
30 | 2, 29 | syl6bi 255 | . . 3 ⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 → (Λ‘𝐴) ≤ (log‘𝐴))) |
31 | 30 | imp 409 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ (Λ‘𝐴) ≠ 0) → (Λ‘𝐴) ≤ (log‘𝐴)) |
32 | nnge1 11659 | . . . 4 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
33 | nnrp 12394 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+) | |
34 | logleb 25184 | . . . . 5 ⊢ ((1 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (1 ≤ 𝐴 ↔ (log‘1) ≤ (log‘𝐴))) | |
35 | 12, 33, 34 | sylancr 589 | . . . 4 ⊢ (𝐴 ∈ ℕ → (1 ≤ 𝐴 ↔ (log‘1) ≤ (log‘𝐴))) |
36 | 32, 35 | mpbid 234 | . . 3 ⊢ (𝐴 ∈ ℕ → (log‘1) ≤ (log‘𝐴)) |
37 | 9, 36 | eqbrtrrid 5095 | . 2 ⊢ (𝐴 ∈ ℕ → 0 ≤ (log‘𝐴)) |
38 | 1, 31, 37 | pm2.61ne 3101 | 1 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) ≤ (log‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∃wrex 3138 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 ℝcr 10529 0cc0 10530 1c1 10531 · cmul 10535 ≤ cle 10669 ℕcn 11631 ℤcz 11975 ℝ+crp 12383 ↑cexp 13426 ℙcprime 16010 logclog 25136 Λcvma 25667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-fi 8868 df-sup 8899 df-inf 8900 df-oi 8967 df-dju 9323 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 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 12890 df-fzo 13031 df-fl 13159 df-mod 13235 df-seq 13367 df-exp 13427 df-fac 13631 df-bc 13660 df-hash 13688 df-shft 14421 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-limsup 14823 df-clim 14840 df-rlim 14841 df-sum 15038 df-ef 15416 df-sin 15418 df-cos 15419 df-pi 15421 df-dvds 15603 df-gcd 15839 df-prm 16011 df-pc 16169 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-hom 16584 df-cco 16585 df-rest 16691 df-topn 16692 df-0g 16710 df-gsum 16711 df-topgen 16712 df-pt 16713 df-prds 16716 df-xrs 16770 df-qtop 16775 df-imas 16776 df-xps 16778 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-mulg 18220 df-cntz 18442 df-cmn 18903 df-psmet 20532 df-xmet 20533 df-met 20534 df-bl 20535 df-mopn 20536 df-fbas 20537 df-fg 20538 df-cnfld 20541 df-top 21497 df-topon 21514 df-topsp 21536 df-bases 21549 df-cld 21622 df-ntr 21623 df-cls 21624 df-nei 21701 df-lp 21739 df-perf 21740 df-cn 21830 df-cnp 21831 df-haus 21918 df-tx 22165 df-hmeo 22358 df-fil 22449 df-fm 22541 df-flim 22542 df-flf 22543 df-xms 22925 df-ms 22926 df-tms 22927 df-cncf 23481 df-limc 24461 df-dv 24462 df-log 25138 df-vma 25673 |
This theorem is referenced by: pntpbnd1a 26159 hgt750lemb 31948 |
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