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Mirrors > Home > MPE Home > Th. List > emcllem4 | Structured version Visualization version GIF version |
Description: Lemma for emcl 25508. The difference between series 𝐹 and 𝐺 tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.) |
Ref | Expression |
---|---|
emcl.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) |
emcl.2 | ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) |
emcl.3 | ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) |
Ref | Expression |
---|---|
emcllem4 | ⊢ 𝐻 ⇝ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 12270 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 12002 | . . 3 ⊢ (⊤ → 1 ∈ ℤ) | |
3 | ax-1cn 10584 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | divcnv 15198 | . . . 4 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0) |
6 | emcl.3 | . . . . 5 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) | |
7 | nnex 11633 | . . . . . 6 ⊢ ℕ ∈ V | |
8 | 7 | mptex 6978 | . . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) ∈ V |
9 | 6, 8 | eqeltri 2909 | . . . 4 ⊢ 𝐻 ∈ V |
10 | 9 | a1i 11 | . . 3 ⊢ (⊤ → 𝐻 ∈ V) |
11 | oveq2 7153 | . . . . . 6 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚)) | |
12 | eqid 2821 | . . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) | |
13 | ovex 7178 | . . . . . 6 ⊢ (1 / 𝑚) ∈ V | |
14 | 11, 12, 13 | fvmpt 6762 | . . . . 5 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚) = (1 / 𝑚)) |
15 | 14 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚) = (1 / 𝑚)) |
16 | nnrecre 11668 | . . . . 5 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ) | |
17 | 16 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ) |
18 | 15, 17 | eqeltrd 2913 | . . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚) ∈ ℝ) |
19 | 11 | oveq2d 7161 | . . . . . . . 8 ⊢ (𝑛 = 𝑚 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑚))) |
20 | 19 | fveq2d 6668 | . . . . . . 7 ⊢ (𝑛 = 𝑚 → (log‘(1 + (1 / 𝑛))) = (log‘(1 + (1 / 𝑚)))) |
21 | fvex 6677 | . . . . . . 7 ⊢ (log‘(1 + (1 / 𝑚))) ∈ V | |
22 | 20, 6, 21 | fvmpt 6762 | . . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝐻‘𝑚) = (log‘(1 + (1 / 𝑚)))) |
23 | 22 | adantl 482 | . . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) = (log‘(1 + (1 / 𝑚)))) |
24 | 1rp 12383 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
25 | nnrp 12390 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+) | |
26 | 25 | adantl 482 | . . . . . . . . 9 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ+) |
27 | 26 | rpreccld 12431 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ+) |
28 | rpaddcl 12401 | . . . . . . . 8 ⊢ ((1 ∈ ℝ+ ∧ (1 / 𝑚) ∈ ℝ+) → (1 + (1 / 𝑚)) ∈ ℝ+) | |
29 | 24, 27, 28 | sylancr 587 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 + (1 / 𝑚)) ∈ ℝ+) |
30 | 29 | rpred 12421 | . . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 + (1 / 𝑚)) ∈ ℝ) |
31 | 1re 10630 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
32 | ltaddrp 12416 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑚) ∈ ℝ+) → 1 < (1 + (1 / 𝑚))) | |
33 | 31, 27, 32 | sylancr 587 | . . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 1 < (1 + (1 / 𝑚))) |
34 | 30, 33 | rplogcld 25139 | . . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ∈ ℝ+) |
35 | 23, 34 | eqeltrd 2913 | . . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℝ+) |
36 | 35 | rpred 12421 | . . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℝ) |
37 | 29 | relogcld 25133 | . . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ∈ ℝ) |
38 | efgt1p 15458 | . . . . . . . 8 ⊢ ((1 / 𝑚) ∈ ℝ+ → (1 + (1 / 𝑚)) < (exp‘(1 / 𝑚))) | |
39 | 27, 38 | syl 17 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 + (1 / 𝑚)) < (exp‘(1 / 𝑚))) |
40 | 17 | rpefcld 15448 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (exp‘(1 / 𝑚)) ∈ ℝ+) |
41 | logltb 25110 | . . . . . . . 8 ⊢ (((1 + (1 / 𝑚)) ∈ ℝ+ ∧ (exp‘(1 / 𝑚)) ∈ ℝ+) → ((1 + (1 / 𝑚)) < (exp‘(1 / 𝑚)) ↔ (log‘(1 + (1 / 𝑚))) < (log‘(exp‘(1 / 𝑚))))) | |
42 | 29, 40, 41 | syl2anc 584 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → ((1 + (1 / 𝑚)) < (exp‘(1 / 𝑚)) ↔ (log‘(1 + (1 / 𝑚))) < (log‘(exp‘(1 / 𝑚))))) |
43 | 39, 42 | mpbid 233 | . . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) < (log‘(exp‘(1 / 𝑚)))) |
44 | 17 | relogefd 25138 | . . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(exp‘(1 / 𝑚))) = (1 / 𝑚)) |
45 | 43, 44 | breqtrd 5084 | . . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) < (1 / 𝑚)) |
46 | 37, 17, 45 | ltled 10777 | . . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ≤ (1 / 𝑚)) |
47 | 46, 23, 15 | 3brtr4d 5090 | . . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚)) |
48 | 35 | rpge0d 12425 | . . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 0 ≤ (𝐻‘𝑚)) |
49 | 1, 2, 5, 10, 18, 36, 47, 48 | climsqz2 14988 | . 2 ⊢ (⊤ → 𝐻 ⇝ 0) |
50 | 49 | mptru 1535 | 1 ⊢ 𝐻 ⇝ 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 Vcvv 3495 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11627 ℝ+crp 12379 ...cfz 12882 ⇝ cli 14831 Σcsu 15032 expce 15405 logclog 25065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 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 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-fi 8864 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 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 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-ioo 12732 df-ioc 12733 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14416 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-limsup 14818 df-clim 14835 df-rlim 14836 df-sum 15033 df-ef 15411 df-sin 15413 df-cos 15414 df-pi 15416 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-rest 16686 df-topn 16687 df-0g 16705 df-gsum 16706 df-topgen 16707 df-pt 16708 df-prds 16711 df-xrs 16765 df-qtop 16770 df-imas 16771 df-xps 16773 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-submnd 17947 df-mulg 18165 df-cntz 18387 df-cmn 18839 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-fbas 20472 df-fg 20473 df-cnfld 20476 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cld 21557 df-ntr 21558 df-cls 21559 df-nei 21636 df-lp 21674 df-perf 21675 df-cn 21765 df-cnp 21766 df-haus 21853 df-tx 22100 df-hmeo 22293 df-fil 22384 df-fm 22476 df-flim 22477 df-flf 22478 df-xms 22859 df-ms 22860 df-tms 22861 df-cncf 23415 df-limc 24393 df-dv 24394 df-log 25067 |
This theorem is referenced by: emcllem6 25506 |
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