emcllem4 - Metamath Proof Explorer

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Theorem emcllem4 26979

Description: Lemma for emcl 26983. 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 12821	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12552	. . 3 ⊢ (⊤ → 1 ∈ ℤ)
3		ax-1cn 11090	. . . 4 ⊢ 1 ∈ ℂ
4		divcnv 15812	. . . 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 12174	. . . . . 6 ⊢ ℕ ∈ V
8	7	mptex 7172	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛)))) ∈ V
9	6, 8	eqeltri 2833	. . . 4 ⊢ 𝐻 ∈ V
10	9	a1i 11	. . 3 ⊢ (⊤ → 𝐻 ∈ V)
11		oveq2 7369	. . . . . 6 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
12		eqid 2737	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
13		ovex 7394	. . . . . 6 ⊢ (1 / 𝑚) ∈ V
14	11, 12, 13	fvmpt 6942	. . . . 5 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚) = (1 / 𝑚))
15	14	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚) = (1 / 𝑚))
16		nnrecre 12213	. . . . 5 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
17	16	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
18	15, 17	eqeltrd 2837	. . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚) ∈ ℝ)
19	11	oveq2d 7377	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑚)))
20	19	fveq2d 6839	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (log‘(1 + (1 / 𝑛))) = (log‘(1 + (1 / 𝑚))))
21		fvex 6848	. . . . . . 7 ⊢ (log‘(1 + (1 / 𝑚))) ∈ V
22	20, 6, 21	fvmpt 6942	. . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝐻‘𝑚) = (log‘(1 + (1 / 𝑚))))
23	22	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) = (log‘(1 + (1 / 𝑚))))
24		1rp 12940	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
25		nnrp 12948	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
26	25	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
27	26	rpreccld 12990	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ⁺)
28		rpaddcl 12960	. . . . . . . 8 ⊢ ((1 ∈ ℝ⁺ ∧ (1 / 𝑚) ∈ ℝ⁺) → (1 + (1 / 𝑚)) ∈ ℝ⁺)
29	24, 27, 28	sylancr 588	. . . . . . 7 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 + (1 / 𝑚)) ∈ ℝ⁺)
30	29	rpred 12980	. . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (1 + (1 / 𝑚)) ∈ ℝ)
31		1re 11138	. . . . . . 7 ⊢ 1 ∈ ℝ
32		ltaddrp 12975	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑚) ∈ ℝ⁺) → 1 < (1 + (1 / 𝑚)))
33	31, 27, 32	sylancr 588	. . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 1 < (1 + (1 / 𝑚)))
34	30, 33	rplogcld 26609	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ∈ ℝ⁺)
35	23, 34	eqeltrd 2837	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℝ⁺)
36	35	rpred 12980	. . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℝ)
37	29	relogcld 26603	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ∈ ℝ)
38		efgt1p 16076	. . . . . . . 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 16066	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (exp‘(1 / 𝑚)) ∈ ℝ⁺)
41		logltb 26580	. . . . . . . 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 585	. . . . . . 7 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → ((1 + (1 / 𝑚)) < (exp‘(1 / 𝑚)) ↔ (log‘(1 + (1 / 𝑚))) < (log‘(exp‘(1 / 𝑚)))))
43	39, 42	mpbid 232	. . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) < (log‘(exp‘(1 / 𝑚))))
44	17	relogefd 26608	. . . . . 6 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(exp‘(1 / 𝑚))) = (1 / 𝑚))
45	43, 44	breqtrd 5112	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) < (1 / 𝑚))
46	37, 17, 45	ltled 11288	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (log‘(1 + (1 / 𝑚))) ≤ (1 / 𝑚))
47	46, 23, 15	3brtr4d 5118	. . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑚))
48	35	rpge0d 12984	. . 3 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ) → 0 ≤ (𝐻‘𝑚))
49	1, 2, 5, 10, 18, 36, 47, 48	climsqz2 15598	. 2 ⊢ (⊤ → 𝐻 ⇝ 0)
50	49	mptru 1549	1 ⊢ 𝐻 ⇝ 0

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 Vcvv 3430 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6493 (class class class)co 7361 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 < clt 11173 ≤ cle 11174 − cmin 11371 / cdiv 11801 ℕcn 12168 ℝ⁺crp 12936 ...cfz 13455 ⇝ cli 15440 Σcsu 15642 expce 16020 logclog 26534

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 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 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-ef 16026 df-sin 16028 df-cos 16029 df-pi 16031 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-limc 25846 df-dv 25847 df-log 26536

This theorem is referenced by: emcllem6 26981

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