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Mirrors > Home > MPE Home > Th. List > harmoniclbnd | 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 |
---|---|
harmoniclbnd | ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relogcl 25159 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
2 | rprege0 12405 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) | |
3 | flge0nn0 13191 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (⌊‘𝐴) ∈ ℕ0) |
5 | nn0p1nn 11937 | . . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ0 → ((⌊‘𝐴) + 1) ∈ ℕ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((⌊‘𝐴) + 1) ∈ ℕ) |
7 | 6 | nnrpd 12430 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((⌊‘𝐴) + 1) ∈ ℝ+) |
8 | relogcl 25159 | . . 3 ⊢ (((⌊‘𝐴) + 1) ∈ ℝ+ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ) |
10 | fzfid 13342 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (1...(⌊‘𝐴)) ∈ Fin) | |
11 | elfznn 12937 | . . . . 5 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ) | |
12 | 11 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ) |
13 | 12 | nnrecred 11689 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ) |
14 | 10, 13 | fsumrecl 15091 | . 2 ⊢ (𝐴 ∈ ℝ+ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ) |
15 | rpre 12398 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
16 | fllep1 13172 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1)) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≤ ((⌊‘𝐴) + 1)) |
18 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+) | |
19 | 18, 7 | logled 25210 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1)))) |
20 | 17, 19 | mpbid 234 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1))) |
21 | harmonicbnd3 25585 | . . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ0 → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ)) | |
22 | 4, 21 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ)) |
23 | 0re 10643 | . . . . . 6 ⊢ 0 ∈ ℝ | |
24 | emre 25583 | . . . . . 6 ⊢ γ ∈ ℝ | |
25 | 23, 24 | elicc2i 12803 | . . . . 5 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ) ↔ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ ∧ 0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∧ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ γ)) |
26 | 25 | simp2bi 1142 | . . . 4 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ) → 0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1)))) |
27 | 22, 26 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1)))) |
28 | 14, 9 | subge0d 11230 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ↔ (log‘((⌊‘𝐴) + 1)) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))) |
29 | 27, 28 | mpbid 234 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘((⌊‘𝐴) + 1)) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) |
30 | 1, 9, 14, 20, 29 | letrd 10797 | 1 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 ≤ cle 10676 − cmin 10870 / cdiv 11297 ℕcn 11638 ℕ0cn0 11898 ℝ+crp 12390 [,]cicc 12742 ...cfz 12893 ⌊cfl 13161 Σcsu 15042 logclog 25138 γcem 25569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-e 15422 df-sin 15423 df-cos 15424 df-tan 15425 df-pi 15426 df-dvds 15608 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-ulm 24965 df-log 25140 df-atan 25445 df-em 25570 |
This theorem is referenced by: fsumharmonic 25589 logfaclbnd 25798 dchrisum0fno1 26087 vmalogdivsum2 26114 pntrsumo1 26141 pntrlog2bndlem6 26159 |
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