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Theorem harmoniclbnd 25187

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

**Assertion**
Ref	Expression
harmoniclbnd	⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))

Distinct variable group: 𝐴,𝑚

**Proof of Theorem harmoniclbnd**
Step	Hyp	Ref	Expression
1		relogcl 24759	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
2		rprege0 12154	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
3		flge0nn0 12940	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
4	2, 3	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
5		nn0p1nn 11683	. . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → ((⌊‘𝐴) + 1) ∈ ℕ)
6	4, 5	syl 17	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℕ)
7	6	nnrpd 12179	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ⁺)
8		relogcl 24759	. . 3 ⊢ (((⌊‘𝐴) + 1) ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
9	7, 8	syl 17	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
10		fzfid 13091	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
11		elfznn 12687	. . . . 5 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
12	11	adantl 475	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
13	12	nnrecred 11426	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
14	10, 13	fsumrecl 14872	. 2 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
15		rpre 12145	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
16		fllep1 12921	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
17	15, 16	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
18		id 22	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
19	18, 7	logled 24810	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1))))
20	17, 19	mpbid 224	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1)))
21		harmonicbnd3 25186	. . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ))
22	4, 21	syl 17	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ))
23		0re 10378	. . . . . 6 ⊢ 0 ∈ ℝ
24		emre 25184	. . . . . 6 ⊢ γ ∈ ℝ
25	23, 24	elicc2i 12551	. . . . 5 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ) ↔ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ ∧ 0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∧ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ γ))
26	25	simp2bi 1137	. . . 4 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ) → 0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
27	22, 26	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
28	14, 9	subge0d 10965	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ↔ (log‘((⌊‘𝐴) + 1)) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
29	27, 28	mpbid 224	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
30	1, 9, 14, 20, 29	letrd 10533	1 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 ≤ cle 10412 − cmin 10606 / cdiv 11032 ℕcn 11374 ℕ₀cn0 11642 ℝ⁺crp 12137 [,]cicc 12490 ...cfz 12643 ⌊cfl 12910 Σcsu 14824 logclog 24738 γcem 25170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-e 15201 df-sin 15202 df-cos 15203 df-pi 15205 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 df-em 25171

This theorem is referenced by: fsumharmonic 25190 logfaclbnd 25399 dchrisum0fno1 25652 vmalogdivsum2 25679 pntrsumo1 25706 pntrlog2bndlem6 25724

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