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Theorem harmoniclbnd 27052

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 26617	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
2		rprege0 13050	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
3		flge0nn0 13860	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
4	2, 3	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
5		nn0p1nn 12565	. . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → ((⌊‘𝐴) + 1) ∈ ℕ)
6	4, 5	syl 17	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℕ)
7	6	nnrpd 13075	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ⁺)
8		relogcl 26617	. . 3 ⊢ (((⌊‘𝐴) + 1) ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
9	7, 8	syl 17	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
10		fzfid 14014	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
11		elfznn 13593	. . . . 5 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
12	11	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
13	12	nnrecred 12317	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
14	10, 13	fsumrecl 15770	. 2 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
15		rpre 13043	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
16		fllep1 13841	. . . 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 26669	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1))))
20	17, 19	mpbid 232	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1)))
21		harmonicbnd3 27051	. . . . 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 11263	. . . . . 6 ⊢ 0 ∈ ℝ
24		emre 27049	. . . . . 6 ⊢ γ ∈ ℝ
25	23, 24	elicc2i 13453	. . . . 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 1147	. . . 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 11853	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ↔ (log‘((⌊‘𝐴) + 1)) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚)))
29	27, 28	mpbid 232	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
30	1, 9, 14, 20, 29	letrd 11418	1 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ₀cn0 12526 ℝ⁺crp 13034 [,]cicc 13390 ...cfz 13547 ⌊cfl 13830 Σcsu 15722 logclog 26596 γcem 27035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-e 16104 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-dvds 16291 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-ulm 26420 df-log 26598 df-atan 26910 df-em 27036

This theorem is referenced by: fsumharmonic 27055 logfaclbnd 27266 dchrisum0fno1 27555 vmalogdivsum2 27582 pntrsumo1 27609 pntrlog2bndlem6 27627

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