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Mirrors > Home > MPE Home > Th. List > harmonicubnd | 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 |
---|---|
harmonicubnd | ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ≤ ((log‘𝐴) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 13870 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin) | |
2 | elfznn 13462 | . . . . 5 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ) | |
3 | 2 | adantl 482 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ) |
4 | 3 | nnrecred 12200 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ) |
5 | 1, 4 | fsumrecl 15611 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ) |
6 | flge1nn 13718 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ) | |
7 | 6 | nnrpd 12947 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℝ+) |
8 | 7 | relogcld 25962 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘(⌊‘𝐴)) ∈ ℝ) |
9 | peano2re 11324 | . . 3 ⊢ ((log‘(⌊‘𝐴)) ∈ ℝ → ((log‘(⌊‘𝐴)) + 1) ∈ ℝ) | |
10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((log‘(⌊‘𝐴)) + 1) ∈ ℝ) |
11 | simpl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ) | |
12 | 0red 11154 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ) | |
13 | 1re 11151 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ) |
15 | 0lt1 11673 | . . . . . . 7 ⊢ 0 < 1 | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 1) |
17 | simpr 485 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴) | |
18 | 12, 14, 11, 16, 17 | ltletrd 11311 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴) |
19 | 11, 18 | elrpd 12946 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+) |
20 | 19 | relogcld 25962 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ) |
21 | peano2re 11324 | . . 3 ⊢ ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ) | |
22 | 20, 21 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((log‘𝐴) + 1) ∈ ℝ) |
23 | harmonicbnd 26337 | . . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ∈ (γ[,]1)) | |
24 | 6, 23 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ∈ (γ[,]1)) |
25 | emre 26339 | . . . . . 6 ⊢ γ ∈ ℝ | |
26 | 25, 13 | elicc2i 13322 | . . . . 5 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ∈ (γ[,]1) ↔ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ∈ ℝ ∧ γ ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ∧ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ≤ 1)) |
27 | 26 | simp3bi 1147 | . . . 4 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ∈ (γ[,]1) → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ≤ 1) |
28 | 24, 27 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ≤ 1) |
29 | 5, 8, 14 | lesubadd2d 11750 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘(⌊‘𝐴))) ≤ 1 ↔ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ≤ ((log‘(⌊‘𝐴)) + 1))) |
30 | 28, 29 | mpbid 231 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ≤ ((log‘(⌊‘𝐴)) + 1)) |
31 | flle 13696 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
32 | 31 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ≤ 𝐴) |
33 | 7, 19 | logled 25966 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) ≤ 𝐴 ↔ (log‘(⌊‘𝐴)) ≤ (log‘𝐴))) |
34 | 32, 33 | mpbid 231 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘(⌊‘𝐴)) ≤ (log‘𝐴)) |
35 | 8, 20, 14, 34 | leadd1dd 11765 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((log‘(⌊‘𝐴)) + 1) ≤ ((log‘𝐴) + 1)) |
36 | 5, 10, 22, 30, 35 | letrd 11308 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ≤ ((log‘𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 ℝcr 11046 0cc0 11047 1c1 11048 + caddc 11050 < clt 11185 ≤ cle 11186 − cmin 11381 / cdiv 11808 ℕcn 12149 [,]cicc 13259 ...cfz 13416 ⌊cfl 13687 Σcsu 15562 logclog 25894 γcem 26325 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-inf2 9573 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 ax-addf 11126 ax-mulf 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-2o 8409 df-er 8644 df-map 8763 df-pm 8764 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-z 12496 df-dec 12615 df-uz 12760 df-q 12866 df-rp 12908 df-xneg 13025 df-xadd 13026 df-xmul 13027 df-ioo 13260 df-ioc 13261 df-ico 13262 df-icc 13263 df-fz 13417 df-fzo 13560 df-fl 13689 df-mod 13767 df-seq 13899 df-exp 13960 df-fac 14166 df-bc 14195 df-hash 14223 df-shft 14944 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-limsup 15345 df-clim 15362 df-rlim 15363 df-sum 15563 df-ef 15942 df-sin 15944 df-cos 15945 df-pi 15947 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-starv 17140 df-sca 17141 df-vsca 17142 df-ip 17143 df-tset 17144 df-ple 17145 df-ds 17147 df-unif 17148 df-hom 17149 df-cco 17150 df-rest 17296 df-topn 17297 df-0g 17315 df-gsum 17316 df-topgen 17317 df-pt 17318 df-prds 17321 df-xrs 17376 df-qtop 17381 df-imas 17382 df-xps 17384 df-mre 17458 df-mrc 17459 df-acs 17461 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-submnd 18594 df-mulg 18864 df-cntz 19088 df-cmn 19555 df-psmet 20773 df-xmet 20774 df-met 20775 df-bl 20776 df-mopn 20777 df-fbas 20778 df-fg 20779 df-cnfld 20782 df-top 22227 df-topon 22244 df-topsp 22266 df-bases 22280 df-cld 22354 df-ntr 22355 df-cls 22356 df-nei 22433 df-lp 22471 df-perf 22472 df-cn 22562 df-cnp 22563 df-haus 22650 df-tx 22897 df-hmeo 23090 df-fil 23181 df-fm 23273 df-flim 23274 df-flf 23275 df-xms 23657 df-ms 23658 df-tms 23659 df-cncf 24225 df-limc 25214 df-dv 25215 df-log 25896 df-em 26326 |
This theorem is referenced by: fsumharmonic 26345 logfaclbnd 26554 vmalogdivsum2 26870 logdivbnd 26888 pntrsumo1 26897 pntrlog2bndlem2 26910 pntrlog2bndlem5 26913 pntrlog2bndlem6 26915 |
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