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Mirrors > Home > MPE Home > Th. List > harmonicbnd3 | Structured version Visualization version GIF version |
Description: A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.) |
Ref | Expression |
---|---|
harmonicbnd3 | ⊢ (𝑁 ∈ ℕ0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11887 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | 0re 10632 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | emre 25591 | . . . . 5 ⊢ γ ∈ ℝ | |
4 | 2re 11699 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
5 | ere 15434 | . . . . . . . . 9 ⊢ e ∈ ℝ | |
6 | egt2lt3 15551 | . . . . . . . . . 10 ⊢ (2 < e ∧ e < 3) | |
7 | 6 | simpli 487 | . . . . . . . . 9 ⊢ 2 < e |
8 | 4, 5, 7 | ltleii 10752 | . . . . . . . 8 ⊢ 2 ≤ e |
9 | 2rp 12382 | . . . . . . . . 9 ⊢ 2 ∈ ℝ+ | |
10 | epr 15553 | . . . . . . . . 9 ⊢ e ∈ ℝ+ | |
11 | logleb 25194 | . . . . . . . . 9 ⊢ ((2 ∈ ℝ+ ∧ e ∈ ℝ+) → (2 ≤ e ↔ (log‘2) ≤ (log‘e))) | |
12 | 9, 10, 11 | mp2an 691 | . . . . . . . 8 ⊢ (2 ≤ e ↔ (log‘2) ≤ (log‘e)) |
13 | 8, 12 | mpbi 233 | . . . . . . 7 ⊢ (log‘2) ≤ (log‘e) |
14 | loge 25178 | . . . . . . 7 ⊢ (log‘e) = 1 | |
15 | 13, 14 | breqtri 5055 | . . . . . 6 ⊢ (log‘2) ≤ 1 |
16 | 1re 10630 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
17 | relogcl 25167 | . . . . . . . 8 ⊢ (2 ∈ ℝ+ → (log‘2) ∈ ℝ) | |
18 | 9, 17 | ax-mp 5 | . . . . . . 7 ⊢ (log‘2) ∈ ℝ |
19 | 16, 18 | subge0i 11182 | . . . . . 6 ⊢ (0 ≤ (1 − (log‘2)) ↔ (log‘2) ≤ 1) |
20 | 15, 19 | mpbir 234 | . . . . 5 ⊢ 0 ≤ (1 − (log‘2)) |
21 | 3 | leidi 11163 | . . . . 5 ⊢ γ ≤ γ |
22 | iccss 12793 | . . . . 5 ⊢ (((0 ∈ ℝ ∧ γ ∈ ℝ) ∧ (0 ≤ (1 − (log‘2)) ∧ γ ≤ γ)) → ((1 − (log‘2))[,]γ) ⊆ (0[,]γ)) | |
23 | 2, 3, 20, 21, 22 | mp4an 692 | . . . 4 ⊢ ((1 − (log‘2))[,]γ) ⊆ (0[,]γ) |
24 | harmonicbnd2 25590 | . . . 4 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ((1 − (log‘2))[,]γ)) | |
25 | 23, 24 | sseldi 3913 | . . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
26 | oveq2 7143 | . . . . . . . . 9 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0)) | |
27 | fz10 12923 | . . . . . . . . 9 ⊢ (1...0) = ∅ | |
28 | 26, 27 | eqtrdi 2849 | . . . . . . . 8 ⊢ (𝑁 = 0 → (1...𝑁) = ∅) |
29 | 28 | sumeq1d 15050 | . . . . . . 7 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = Σ𝑚 ∈ ∅ (1 / 𝑚)) |
30 | sum0 15070 | . . . . . . 7 ⊢ Σ𝑚 ∈ ∅ (1 / 𝑚) = 0 | |
31 | 29, 30 | eqtrdi 2849 | . . . . . 6 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = 0) |
32 | fv0p1e1 11748 | . . . . . . 7 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = (log‘1)) | |
33 | log1 25177 | . . . . . . 7 ⊢ (log‘1) = 0 | |
34 | 32, 33 | eqtrdi 2849 | . . . . . 6 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = 0) |
35 | 31, 34 | oveq12d 7153 | . . . . 5 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = (0 − 0)) |
36 | 0m0e0 11745 | . . . . 5 ⊢ (0 − 0) = 0 | |
37 | 35, 36 | eqtrdi 2849 | . . . 4 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = 0) |
38 | 2 | leidi 11163 | . . . . 5 ⊢ 0 ≤ 0 |
39 | emgt0 25592 | . . . . . 6 ⊢ 0 < γ | |
40 | 2, 3, 39 | ltleii 10752 | . . . . 5 ⊢ 0 ≤ γ |
41 | 2, 3 | elicc2i 12791 | . . . . 5 ⊢ (0 ∈ (0[,]γ) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ γ)) |
42 | 2, 38, 40, 41 | mpbir3an 1338 | . . . 4 ⊢ 0 ∈ (0[,]γ) |
43 | 37, 42 | eqeltrdi 2898 | . . 3 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
44 | 25, 43 | jaoi 854 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
45 | 1, 44 | sylbi 220 | 1 ⊢ (𝑁 ∈ ℕ0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 3c3 11681 ℕ0cn0 11885 ℝ+crp 12377 [,]cicc 12729 ...cfz 12885 Σcsu 15034 eceu 15408 logclog 25146 γcem 25577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-e 15414 df-sin 15415 df-cos 15416 df-tan 15417 df-pi 15418 df-dvds 15600 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-ulm 24972 df-log 25148 df-atan 25453 df-em 25578 |
This theorem is referenced by: harmoniclbnd 25594 harmonicbnd4 25596 logdivbnd 26140 |
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