harmonicbnd3 - Metamath Proof Explorer

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Theorem harmonicbnd3 26166

Description: A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

**Assertion**
Ref	Expression
harmonicbnd3	⊢ (𝑁 ∈ ℕ₀ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))

Distinct variable group: 𝑚,𝑁

**Proof of Theorem harmonicbnd3**
Step	Hyp	Ref	Expression
1		elnn0 12244	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		0re 10986	. . . . 5 ⊢ 0 ∈ ℝ
3		emre 26164	. . . . 5 ⊢ γ ∈ ℝ
4		2re 12056	. . . . . . . . 9 ⊢ 2 ∈ ℝ
5		ere 15807	. . . . . . . . 9 ⊢ e ∈ ℝ
6		egt2lt3 15924	. . . . . . . . . 10 ⊢ (2 < e ∧ e < 3)
7	6	simpli 484	. . . . . . . . 9 ⊢ 2 < e
8	4, 5, 7	ltleii 11107	. . . . . . . 8 ⊢ 2 ≤ e
9		2rp 12744	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
10		epr 15926	. . . . . . . . 9 ⊢ e ∈ ℝ⁺
11		logleb 25767	. . . . . . . . 9 ⊢ ((2 ∈ ℝ⁺ ∧ e ∈ ℝ⁺) → (2 ≤ e ↔ (log‘2) ≤ (log‘e)))
12	9, 10, 11	mp2an 689	. . . . . . . 8 ⊢ (2 ≤ e ↔ (log‘2) ≤ (log‘e))
13	8, 12	mpbi 229	. . . . . . 7 ⊢ (log‘2) ≤ (log‘e)
14		loge 25751	. . . . . . 7 ⊢ (log‘e) = 1
15	13, 14	breqtri 5100	. . . . . 6 ⊢ (log‘2) ≤ 1
16		1re 10984	. . . . . . 7 ⊢ 1 ∈ ℝ
17		relogcl 25740	. . . . . . . 8 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
18	9, 17	ax-mp 5	. . . . . . 7 ⊢ (log‘2) ∈ ℝ
19	16, 18	subge0i 11537	. . . . . 6 ⊢ (0 ≤ (1 − (log‘2)) ↔ (log‘2) ≤ 1)
20	15, 19	mpbir 230	. . . . 5 ⊢ 0 ≤ (1 − (log‘2))
21	3	leidi 11518	. . . . 5 ⊢ γ ≤ γ
22		iccss 13156	. . . . 5 ⊢ (((0 ∈ ℝ ∧ γ ∈ ℝ) ∧ (0 ≤ (1 − (log‘2)) ∧ γ ≤ γ)) → ((1 − (log‘2))[,]γ) ⊆ (0[,]γ))
23	2, 3, 20, 21, 22	mp4an 690	. . . 4 ⊢ ((1 − (log‘2))[,]γ) ⊆ (0[,]γ)
24		harmonicbnd2 26163	. . . 4 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ((1 − (log‘2))[,]γ))
25	23, 24	sselid 3920	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
26		oveq2 7292	. . . . . . . . 9 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0))
27		fz10 13286	. . . . . . . . 9 ⊢ (1...0) = ∅
28	26, 27	eqtrdi 2795	. . . . . . . 8 ⊢ (𝑁 = 0 → (1...𝑁) = ∅)
29	28	sumeq1d 15422	. . . . . . 7 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = Σ𝑚 ∈ ∅ (1 / 𝑚))
30		sum0 15442	. . . . . . 7 ⊢ Σ𝑚 ∈ ∅ (1 / 𝑚) = 0
31	29, 30	eqtrdi 2795	. . . . . 6 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = 0)
32		fv0p1e1 12105	. . . . . . 7 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = (log‘1))
33		log1 25750	. . . . . . 7 ⊢ (log‘1) = 0
34	32, 33	eqtrdi 2795	. . . . . 6 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = 0)
35	31, 34	oveq12d 7302	. . . . 5 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = (0 − 0))
36		0m0e0 12102	. . . . 5 ⊢ (0 − 0) = 0
37	35, 36	eqtrdi 2795	. . . 4 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = 0)
38	2	leidi 11518	. . . . 5 ⊢ 0 ≤ 0
39		emgt0 26165	. . . . . 6 ⊢ 0 < γ
40	2, 3, 39	ltleii 11107	. . . . 5 ⊢ 0 ≤ γ
41	2, 3	elicc2i 13154	. . . . 5 ⊢ (0 ∈ (0[,]γ) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ γ))
42	2, 38, 40, 41	mpbir3an 1340	. . . 4 ⊢ 0 ∈ (0[,]γ)
43	37, 42	eqeltrdi 2848	. . 3 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
44	25, 43	jaoi 854	. 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
45	1, 44	sylbi 216	1 ⊢ (𝑁 ∈ ℕ₀ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∨ wo 844 = wceq 1539 ∈ wcel 2107 ⊆ wss 3888 ∅c0 4257 class class class wbr 5075 ‘cfv 6437 (class class class)co 7284 ℝcr 10879 0cc0 10880 1c1 10881 + caddc 10883 < clt 11018 ≤ cle 11019 − cmin 11214 / cdiv 11641 ℕcn 11982 2c2 12037 3c3 12038 ℕ₀cn0 12242 ℝ⁺crp 12739 [,]cicc 13091 ...cfz 13248 Σcsu 15406 eceu 15781 logclog 25719 γcem 26150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-inf2 9408 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 ax-pre-sup 10958 ax-addf 10959 ax-mulf 10960

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rmo 3072 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-isom 6446 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-of 7542 df-om 7722 df-1st 7840 df-2nd 7841 df-supp 7987 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-2o 8307 df-oadd 8310 df-er 8507 df-map 8626 df-pm 8627 df-ixp 8695 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-fsupp 9138 df-fi 9179 df-sup 9210 df-inf 9211 df-oi 9278 df-card 9706 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-div 11642 df-nn 11983 df-2 12045 df-3 12046 df-4 12047 df-5 12048 df-6 12049 df-7 12050 df-8 12051 df-9 12052 df-n0 12243 df-xnn0 12315 df-z 12329 df-dec 12447 df-uz 12592 df-q 12698 df-rp 12740 df-xneg 12857 df-xadd 12858 df-xmul 12859 df-ioo 13092 df-ioc 13093 df-ico 13094 df-icc 13095 df-fz 13249 df-fzo 13392 df-fl 13521 df-mod 13599 df-seq 13731 df-exp 13792 df-fac 13997 df-bc 14026 df-hash 14054 df-shft 14787 df-cj 14819 df-re 14820 df-im 14821 df-sqrt 14955 df-abs 14956 df-limsup 15189 df-clim 15206 df-rlim 15207 df-sum 15407 df-ef 15786 df-e 15787 df-sin 15788 df-cos 15789 df-tan 15790 df-pi 15791 df-dvds 15973 df-struct 16857 df-sets 16874 df-slot 16892 df-ndx 16904 df-base 16922 df-ress 16951 df-plusg 16984 df-mulr 16985 df-starv 16986 df-sca 16987 df-vsca 16988 df-ip 16989 df-tset 16990 df-ple 16991 df-ds 16993 df-unif 16994 df-hom 16995 df-cco 16996 df-rest 17142 df-topn 17143 df-0g 17161 df-gsum 17162 df-topgen 17163 df-pt 17164 df-prds 17167 df-xrs 17222 df-qtop 17227 df-imas 17228 df-xps 17230 df-mre 17304 df-mrc 17305 df-acs 17307 df-mgm 18335 df-sgrp 18384 df-mnd 18395 df-submnd 18440 df-mulg 18710 df-cntz 18932 df-cmn 19397 df-psmet 20598 df-xmet 20599 df-met 20600 df-bl 20601 df-mopn 20602 df-fbas 20603 df-fg 20604 df-cnfld 20607 df-top 22052 df-topon 22069 df-topsp 22091 df-bases 22105 df-cld 22179 df-ntr 22180 df-cls 22181 df-nei 22258 df-lp 22296 df-perf 22297 df-cn 22387 df-cnp 22388 df-haus 22475 df-cmp 22547 df-tx 22722 df-hmeo 22915 df-fil 23006 df-fm 23098 df-flim 23099 df-flf 23100 df-xms 23482 df-ms 23483 df-tms 23484 df-cncf 24050 df-limc 25039 df-dv 25040 df-ulm 25545 df-log 25721 df-atan 26026 df-em 26151

This theorem is referenced by: harmoniclbnd 26167 harmonicbnd4 26169 logdivbnd 26713

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