harmonicbnd3 - Metamath Proof Explorer

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Theorem harmonicbnd3 26749

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 12479	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		0re 11221	. . . . 5 ⊢ 0 ∈ ℝ
3		emre 26747	. . . . 5 ⊢ γ ∈ ℝ
4		2re 12291	. . . . . . . . 9 ⊢ 2 ∈ ℝ
5		ere 16037	. . . . . . . . 9 ⊢ e ∈ ℝ
6		egt2lt3 16154	. . . . . . . . . 10 ⊢ (2 < e ∧ e < 3)
7	6	simpli 483	. . . . . . . . 9 ⊢ 2 < e
8	4, 5, 7	ltleii 11342	. . . . . . . 8 ⊢ 2 ≤ e
9		2rp 12984	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
10		epr 16156	. . . . . . . . 9 ⊢ e ∈ ℝ⁺
11		logleb 26348	. . . . . . . . 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 26332	. . . . . . 7 ⊢ (log‘e) = 1
15	13, 14	breqtri 5173	. . . . . 6 ⊢ (log‘2) ≤ 1
16		1re 11219	. . . . . . 7 ⊢ 1 ∈ ℝ
17		relogcl 26321	. . . . . . . 8 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
18	9, 17	ax-mp 5	. . . . . . 7 ⊢ (log‘2) ∈ ℝ
19	16, 18	subge0i 11772	. . . . . 6 ⊢ (0 ≤ (1 − (log‘2)) ↔ (log‘2) ≤ 1)
20	15, 19	mpbir 230	. . . . 5 ⊢ 0 ≤ (1 − (log‘2))
21	3	leidi 11753	. . . . 5 ⊢ γ ≤ γ
22		iccss 13397	. . . . 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 26746	. . . 4 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ((1 − (log‘2))[,]γ))
25	23, 24	sselid 3980	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
26		oveq2 7420	. . . . . . . . 9 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0))
27		fz10 13527	. . . . . . . . 9 ⊢ (1...0) = ∅
28	26, 27	eqtrdi 2787	. . . . . . . 8 ⊢ (𝑁 = 0 → (1...𝑁) = ∅)
29	28	sumeq1d 15652	. . . . . . 7 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = Σ𝑚 ∈ ∅ (1 / 𝑚))
30		sum0 15672	. . . . . . 7 ⊢ Σ𝑚 ∈ ∅ (1 / 𝑚) = 0
31	29, 30	eqtrdi 2787	. . . . . 6 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = 0)
32		fv0p1e1 12340	. . . . . . 7 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = (log‘1))
33		log1 26331	. . . . . . 7 ⊢ (log‘1) = 0
34	32, 33	eqtrdi 2787	. . . . . 6 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = 0)
35	31, 34	oveq12d 7430	. . . . 5 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = (0 − 0))
36		0m0e0 12337	. . . . 5 ⊢ (0 − 0) = 0
37	35, 36	eqtrdi 2787	. . . 4 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = 0)
38	2	leidi 11753	. . . . 5 ⊢ 0 ≤ 0
39		emgt0 26748	. . . . . 6 ⊢ 0 < γ
40	2, 3, 39	ltleii 11342	. . . . 5 ⊢ 0 ≤ γ
41	2, 3	elicc2i 13395	. . . . 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 2840	. . 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 1540 ∈ wcel 2105 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℝcr 11113 0cc0 11114 1c1 11115 + caddc 11117 < clt 11253 ≤ cle 11254 − cmin 11449 / cdiv 11876 ℕcn 12217 2c2 12272 3c3 12273 ℕ₀cn0 12477 ℝ⁺crp 12979 [,]cicc 13332 ...cfz 13489 Σcsu 15637 eceu 16011 logclog 26300 γcem 26733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-xnn0 12550 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-e 16017 df-sin 16018 df-cos 16019 df-tan 16020 df-pi 16021 df-dvds 16203 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-cmp 23112 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617 df-ulm 26126 df-log 26302 df-atan 26609 df-em 26734

This theorem is referenced by: harmoniclbnd 26750 harmonicbnd4 26752 logdivbnd 27296

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