harmonicbnd3 - Metamath Proof Explorer

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Theorem harmonicbnd3 25585

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 11900	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		0re 10643	. . . . 5 ⊢ 0 ∈ ℝ
3		emre 25583	. . . . 5 ⊢ γ ∈ ℝ
4		2re 11712	. . . . . . . . 9 ⊢ 2 ∈ ℝ
5		ere 15442	. . . . . . . . 9 ⊢ e ∈ ℝ
6		egt2lt3 15559	. . . . . . . . . 10 ⊢ (2 < e ∧ e < 3)
7	6	simpli 486	. . . . . . . . 9 ⊢ 2 < e
8	4, 5, 7	ltleii 10763	. . . . . . . 8 ⊢ 2 ≤ e
9		2rp 12395	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
10		epr 15561	. . . . . . . . 9 ⊢ e ∈ ℝ⁺
11		logleb 25186	. . . . . . . . 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 232	. . . . . . 7 ⊢ (log‘2) ≤ (log‘e)
14		loge 25170	. . . . . . 7 ⊢ (log‘e) = 1
15	13, 14	breqtri 5091	. . . . . 6 ⊢ (log‘2) ≤ 1
16		1re 10641	. . . . . . 7 ⊢ 1 ∈ ℝ
17		relogcl 25159	. . . . . . . 8 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
18	9, 17	ax-mp 5	. . . . . . 7 ⊢ (log‘2) ∈ ℝ
19	16, 18	subge0i 11193	. . . . . 6 ⊢ (0 ≤ (1 − (log‘2)) ↔ (log‘2) ≤ 1)
20	15, 19	mpbir 233	. . . . 5 ⊢ 0 ≤ (1 − (log‘2))
21	3	leidi 11174	. . . . 5 ⊢ γ ≤ γ
22		iccss 12805	. . . . 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 25582	. . . 4 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ((1 − (log‘2))[,]γ))
25	23, 24	sseldi 3965	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
26		oveq2 7164	. . . . . . . . 9 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0))
27		fz10 12929	. . . . . . . . 9 ⊢ (1...0) = ∅
28	26, 27	syl6eq 2872	. . . . . . . 8 ⊢ (𝑁 = 0 → (1...𝑁) = ∅)
29	28	sumeq1d 15058	. . . . . . 7 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = Σ𝑚 ∈ ∅ (1 / 𝑚))
30		sum0 15078	. . . . . . 7 ⊢ Σ𝑚 ∈ ∅ (1 / 𝑚) = 0
31	29, 30	syl6eq 2872	. . . . . 6 ⊢ (𝑁 = 0 → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) = 0)
32		fv0p1e1 11761	. . . . . . 7 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = (log‘1))
33		log1 25169	. . . . . . 7 ⊢ (log‘1) = 0
34	32, 33	syl6eq 2872	. . . . . 6 ⊢ (𝑁 = 0 → (log‘(𝑁 + 1)) = 0)
35	31, 34	oveq12d 7174	. . . . 5 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = (0 − 0))
36		0m0e0 11758	. . . . 5 ⊢ (0 − 0) = 0
37	35, 36	syl6eq 2872	. . . 4 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) = 0)
38	2	leidi 11174	. . . . 5 ⊢ 0 ≤ 0
39		emgt0 25584	. . . . . 6 ⊢ 0 < γ
40	2, 3, 39	ltleii 10763	. . . . 5 ⊢ 0 ≤ γ
41	2, 3	elicc2i 12803	. . . . 5 ⊢ (0 ∈ (0[,]γ) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ γ))
42	2, 38, 40, 41	mpbir3an 1337	. . . 4 ⊢ 0 ∈ (0[,]γ)
43	37, 42	eqeltrdi 2921	. . 3 ⊢ (𝑁 = 0 → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
44	25, 43	jaoi 853	. 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))
45	1, 44	sylbi 219	1 ⊢ (𝑁 ∈ ℕ₀ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ (0[,]γ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 ℕcn 11638 2c2 11693 3c3 11694 ℕ₀cn0 11898 ℝ⁺crp 12390 [,]cicc 12742 ...cfz 12893 Σcsu 15042 eceu 15416 logclog 25138 γcem 25569

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-e 15422 df-sin 15423 df-cos 15424 df-tan 15425 df-pi 15426 df-dvds 15608 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-ulm 24965 df-log 25140 df-atan 25445 df-em 25570

This theorem is referenced by: harmoniclbnd 25586 harmonicbnd4 25588 logdivbnd 26132

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