harmonicbnd3 - Metamath Proof Explorer

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Theorem harmonicbnd3 27036

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 12526	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		0re 11266	. . . . 5 ⊢ 0 ∈ ℝ
3		emre 27034	. . . . 5 ⊢ γ ∈ ℝ
4		2re 12338	. . . . . . . . 9 ⊢ 2 ∈ ℝ
5		ere 16091	. . . . . . . . 9 ⊢ e ∈ ℝ
6		egt2lt3 16208	. . . . . . . . . 10 ⊢ (2 < e ∧ e < 3)
7	6	simpli 482	. . . . . . . . 9 ⊢ 2 < e
8	4, 5, 7	ltleii 11387	. . . . . . . 8 ⊢ 2 ≤ e
9		2rp 13033	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
10		epr 16210	. . . . . . . . 9 ⊢ e ∈ ℝ⁺
11		logleb 26630	. . . . . . . . 9 ⊢ ((2 ∈ ℝ⁺ ∧ e ∈ ℝ⁺) → (2 ≤ e ↔ (log‘2) ≤ (log‘e)))
12	9, 10, 11	mp2an 690	. . . . . . . 8 ⊢ (2 ≤ e ↔ (log‘2) ≤ (log‘e))
13	8, 12	mpbi 229	. . . . . . 7 ⊢ (log‘2) ≤ (log‘e)
14		loge 26613	. . . . . . 7 ⊢ (log‘e) = 1
15	13, 14	breqtri 5178	. . . . . 6 ⊢ (log‘2) ≤ 1
16		1re 11264	. . . . . . 7 ⊢ 1 ∈ ℝ
17		relogcl 26602	. . . . . . . 8 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
18	9, 17	ax-mp 5	. . . . . . 7 ⊢ (log‘2) ∈ ℝ
19	16, 18	subge0i 11817	. . . . . 6 ⊢ (0 ≤ (1 − (log‘2)) ↔ (log‘2) ≤ 1)
20	15, 19	mpbir 230	. . . . 5 ⊢ 0 ≤ (1 − (log‘2))
21	3	leidi 11798	. . . . 5 ⊢ γ ≤ γ
22		iccss 13446	. . . . 5 ⊢ (((0 ∈ ℝ ∧ γ ∈ ℝ) ∧ (0 ≤ (1 − (log‘2)) ∧ γ ≤ γ)) → ((1 − (log‘2))[,]γ) ⊆ (0[,]γ))
23	2, 3, 20, 21, 22	mp4an 691	. . . 4 ⊢ ((1 − (log‘2))[,]γ) ⊆ (0[,]γ)
24		harmonicbnd2 27033	. . . 4 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ((1 − (log‘2))[,]γ))
25	23, 24	sselid 3977	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
26		oveq2 7432	. . . . . . . . 9 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0))
27		fz10 13576	. . . . . . . . 9 ⊢ (1...0) = ∅
28	26, 27	eqtrdi 2782	. . . . . . . 8 ⊢ (𝑁 = 0 → (1...𝑁) = ∅)
29	28	sumeq1d 15705	. . . . . . 7 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = Σ𝑚 ∈ ∅ (1 / 𝑚))
30		sum0 15725	. . . . . . 7 ⊢ Σ𝑚 ∈ ∅ (1 / 𝑚) = 0
31	29, 30	eqtrdi 2782	. . . . . 6 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = 0)
32		fv0p1e1 12387	. . . . . . 7 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = (log‘1))
33		log1 26612	. . . . . . 7 ⊢ (log‘1) = 0
34	32, 33	eqtrdi 2782	. . . . . 6 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = 0)
35	31, 34	oveq12d 7442	. . . . 5 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = (0 − 0))
36		0m0e0 12384	. . . . 5 ⊢ (0 − 0) = 0
37	35, 36	eqtrdi 2782	. . . 4 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = 0)
38	2	leidi 11798	. . . . 5 ⊢ 0 ≤ 0
39		emgt0 27035	. . . . . 6 ⊢ 0 < γ
40	2, 3, 39	ltleii 11387	. . . . 5 ⊢ 0 ≤ γ
41	2, 3	elicc2i 13444	. . . . 5 ⊢ (0 ∈ (0[,]γ) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ γ))
42	2, 38, 40, 41	mpbir3an 1338	. . . 4 ⊢ 0 ∈ (0[,]γ)
43	37, 42	eqeltrdi 2834	. . 3 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
44	25, 43	jaoi 855	. 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 845 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ∅c0 4325 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 < clt 11298 ≤ cle 11299 − cmin 11494 / cdiv 11921 ℕcn 12264 2c2 12319 3c3 12320 ℕ₀cn0 12524 ℝ⁺crp 13028 [,]cicc 13381 ...cfz 13538 Σcsu 15690 eceu 16064 logclog 26581 γcem 27020

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12597 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-ef 16069 df-e 16070 df-sin 16071 df-cos 16072 df-tan 16073 df-pi 16074 df-dvds 16257 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-cmp 23382 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cncf 24889 df-limc 25886 df-dv 25887 df-ulm 26406 df-log 26583 df-atan 26895 df-em 27021

This theorem is referenced by: harmoniclbnd 27037 harmonicbnd4 27039 logdivbnd 27585

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