emcllem4 - Metamath Proof Explorer

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Theorem emcllem4 24725

Description: Lemma for emcl 24729. The difference between series 𝐹 and 𝐺 tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)

**Hypotheses**
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))))

**Assertion**
Ref	Expression
emcllem4	⊢ 𝐻 ⇝ 0

Distinct variable groups: 𝑚,𝐻 𝑚,𝑛 Allowed substitution hints: 𝐹(𝑚,𝑛) 𝐺(𝑚,𝑛) 𝐻(𝑛)

**Proof of Theorem emcllem4**
Step	Hyp	Ref	Expression
1		nnuz 11723	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 11408	. . 3 ⊢ (⊤ → 1 ∈ ℤ)
3		ax-1cn 9994	. . . 4 ⊢ 1 ∈ ℂ
4		divcnv 14585	. . . 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 11026	. . . . . 6 ⊢ ℕ ∈ V
8	7	mptex 6486	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) ∈ V
9	6, 8	eqeltri 2697	. . . 4 ⊢ 𝐻 ∈ V
10	9	a1i 11	. . 3 ⊢ (⊤ → 𝐻 ∈ V)
11		oveq2 6658	. . . . . 6 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
12		eqid 2622	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
13		ovex 6678	. . . . . 6 ⊢ (1 / 𝑚) ∈ V
14	11, 12, 13	fvmpt 6282	. . . . 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 11057	. . . . 5 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
17	16	adantl 482	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
18	15, 17	eqeltrd 2701	. . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚) ∈ ℝ)
19	11	oveq2d 6666	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑚)))
20	19	fveq2d 6195	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (log‘(1 + (1 / 𝑛))) = (log‘(1 + (1 / 𝑚))))
21		fvex 6201	. . . . . . 7 ⊢ (log‘(1 + (1 / 𝑚))) ∈ V
22	20, 6, 21	fvmpt 6282	. . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝐻‘𝑚) = (log‘(1 + (1 / 𝑚))))
23	22	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) = (log‘(1 + (1 / 𝑚))))
24		1rp 11836	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
25		nnrp 11842	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
26	25	adantl 482	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
27	26	rpreccld 11882	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ⁺)
28		rpaddcl 11854	. . . . . . . 8 ⊢ ((1 ∈ ℝ⁺ ∧ (1 / 𝑚) ∈ ℝ⁺) → (1 + (1 / 𝑚)) ∈ ℝ⁺)
29	24, 27, 28	sylancr 695	. . . . . . 7 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 + (1 / 𝑚)) ∈ ℝ⁺)
30	29	rpred 11872	. . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 + (1 / 𝑚)) ∈ ℝ)
31		1re 10039	. . . . . . 7 ⊢ 1 ∈ ℝ
32		ltaddrp 11867	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑚) ∈ ℝ⁺) → 1 < (1 + (1 / 𝑚)))
33	31, 27, 32	sylancr 695	. . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 1 < (1 + (1 / 𝑚)))
34	30, 33	rplogcld 24375	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ∈ ℝ⁺)
35	23, 34	eqeltrd 2701	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℝ⁺)
36	35	rpred 11872	. . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℝ)
37	29	relogcld 24369	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ∈ ℝ)
38		efgt1p 14845	. . . . . . . 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 14835	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (exp‘(1 / 𝑚)) ∈ ℝ⁺)
41		logltb 24346	. . . . . . . 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 693	. . . . . . 7 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → ((1 + (1 / 𝑚)) < (exp‘(1 / 𝑚)) ↔ (log‘(1 + (1 / 𝑚))) < (log‘(exp‘(1 / 𝑚)))))
43	39, 42	mpbid 222	. . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) < (log‘(exp‘(1 / 𝑚))))
44	17	relogefd 24374	. . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(exp‘(1 / 𝑚))) = (1 / 𝑚))
45	43, 44	breqtrd 4679	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) < (1 / 𝑚))
46	37, 17, 45	ltled 10185	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ≤ (1 / 𝑚))
47	46, 23, 15	3brtr4d 4685	. . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚))
48	35	rpge0d 11876	. . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 0 ≤ (𝐻‘𝑚))
49	1, 2, 5, 10, 18, 36, 47, 48	climsqz2 14372	. 2 ⊢ (⊤ → 𝐻 ⇝ 0)
50	49	trud 1493	1 ⊢ 𝐻 ⇝ 0

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℝ⁺crp 11832 ...cfz 12326 ⇝ cli 14215 Σcsu 14416 expce 14792 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303

This theorem is referenced by: emcllem6 24727

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