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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 11486 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | 0re 10232 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | emre 24931 | . . . . 5 ⊢ γ ∈ ℝ | |
4 | 2re 11282 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
5 | ere 15018 | . . . . . . . . 9 ⊢ e ∈ ℝ | |
6 | egt2lt3 15133 | . . . . . . . . . 10 ⊢ (2 < e ∧ e < 3) | |
7 | 6 | simpli 476 | . . . . . . . . 9 ⊢ 2 < e |
8 | 4, 5, 7 | ltleii 10352 | . . . . . . . 8 ⊢ 2 ≤ e |
9 | 2rp 12030 | . . . . . . . . 9 ⊢ 2 ∈ ℝ+ | |
10 | epr 15135 | . . . . . . . . 9 ⊢ e ∈ ℝ+ | |
11 | logleb 24548 | . . . . . . . . 9 ⊢ ((2 ∈ ℝ+ ∧ e ∈ ℝ+) → (2 ≤ e ↔ (log‘2) ≤ (log‘e))) | |
12 | 9, 10, 11 | mp2an 710 | . . . . . . . 8 ⊢ (2 ≤ e ↔ (log‘2) ≤ (log‘e)) |
13 | 8, 12 | mpbi 220 | . . . . . . 7 ⊢ (log‘2) ≤ (log‘e) |
14 | loge 24532 | . . . . . . 7 ⊢ (log‘e) = 1 | |
15 | 13, 14 | breqtri 4829 | . . . . . 6 ⊢ (log‘2) ≤ 1 |
16 | 1re 10231 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
17 | relogcl 24521 | . . . . . . . 8 ⊢ (2 ∈ ℝ+ → (log‘2) ∈ ℝ) | |
18 | 9, 17 | ax-mp 5 | . . . . . . 7 ⊢ (log‘2) ∈ ℝ |
19 | 16, 18 | subge0i 10773 | . . . . . 6 ⊢ (0 ≤ (1 − (log‘2)) ↔ (log‘2) ≤ 1) |
20 | 15, 19 | mpbir 221 | . . . . 5 ⊢ 0 ≤ (1 − (log‘2)) |
21 | 3 | leidi 10754 | . . . . 5 ⊢ γ ≤ γ |
22 | iccss 12434 | . . . . 5 ⊢ (((0 ∈ ℝ ∧ γ ∈ ℝ) ∧ (0 ≤ (1 − (log‘2)) ∧ γ ≤ γ)) → ((1 − (log‘2))[,]γ) ⊆ (0[,]γ)) | |
23 | 2, 3, 20, 21, 22 | mp4an 711 | . . . 4 ⊢ ((1 − (log‘2))[,]γ) ⊆ (0[,]γ) |
24 | harmonicbnd2 24930 | . . . 4 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ((1 − (log‘2))[,]γ)) | |
25 | 23, 24 | sseldi 3742 | . . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
26 | oveq2 6821 | . . . . . . . . 9 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0)) | |
27 | fz10 12555 | . . . . . . . . 9 ⊢ (1...0) = ∅ | |
28 | 26, 27 | syl6eq 2810 | . . . . . . . 8 ⊢ (𝑁 = 0 → (1...𝑁) = ∅) |
29 | 28 | sumeq1d 14630 | . . . . . . 7 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = Σ𝑚 ∈ ∅ (1 / 𝑚)) |
30 | sum0 14651 | . . . . . . 7 ⊢ Σ𝑚 ∈ ∅ (1 / 𝑚) = 0 | |
31 | 29, 30 | syl6eq 2810 | . . . . . 6 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = 0) |
32 | oveq1 6820 | . . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
33 | 0p1e1 11324 | . . . . . . . . 9 ⊢ (0 + 1) = 1 | |
34 | 32, 33 | syl6eq 2810 | . . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1) |
35 | 34 | fveq2d 6356 | . . . . . . 7 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = (log‘1)) |
36 | log1 24531 | . . . . . . 7 ⊢ (log‘1) = 0 | |
37 | 35, 36 | syl6eq 2810 | . . . . . 6 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = 0) |
38 | 31, 37 | oveq12d 6831 | . . . . 5 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = (0 − 0)) |
39 | 0m0e0 11322 | . . . . 5 ⊢ (0 − 0) = 0 | |
40 | 38, 39 | syl6eq 2810 | . . . 4 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = 0) |
41 | 2 | leidi 10754 | . . . . 5 ⊢ 0 ≤ 0 |
42 | emgt0 24932 | . . . . . 6 ⊢ 0 < γ | |
43 | 2, 3, 42 | ltleii 10352 | . . . . 5 ⊢ 0 ≤ γ |
44 | 2, 3 | elicc2i 12432 | . . . . 5 ⊢ (0 ∈ (0[,]γ) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ γ)) |
45 | 2, 41, 43, 44 | mpbir3an 1427 | . . . 4 ⊢ 0 ∈ (0[,]γ) |
46 | 40, 45 | syl6eqel 2847 | . . 3 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
47 | 25, 46 | jaoi 393 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
48 | 1, 47 | sylbi 207 | 1 ⊢ (𝑁 ∈ ℕ0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 ∅c0 4058 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 ℝcr 10127 0cc0 10128 1c1 10129 + caddc 10131 < clt 10266 ≤ cle 10267 − cmin 10458 / cdiv 10876 ℕcn 11212 2c2 11262 3c3 11263 ℕ0cn0 11484 ℝ+crp 12025 [,]cicc 12371 ...cfz 12519 Σcsu 14615 eceu 14992 logclog 24500 γcem 24917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ioc 12373 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-mod 12863 df-seq 12996 df-exp 13055 df-fac 13255 df-bc 13284 df-hash 13312 df-shft 14006 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-limsup 14401 df-clim 14418 df-rlim 14419 df-sum 14616 df-ef 14997 df-e 14998 df-sin 14999 df-cos 15000 df-pi 15002 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-pt 16307 df-prds 16310 df-xrs 16364 df-qtop 16369 df-imas 16370 df-xps 16372 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-fbas 19945 df-fg 19946 df-cnfld 19949 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 df-nei 21104 df-lp 21142 df-perf 21143 df-cn 21233 df-cnp 21234 df-haus 21321 df-tx 21567 df-hmeo 21760 df-fil 21851 df-fm 21943 df-flim 21944 df-flf 21945 df-xms 22326 df-ms 22327 df-tms 22328 df-cncf 22882 df-limc 23829 df-dv 23830 df-log 24502 df-em 24918 |
This theorem is referenced by: harmoniclbnd 24934 harmonicbnd4 24936 logdivbnd 25444 |
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