divlogrlim - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > divlogrlim		Structured version Visualization version GIF version

Theorem divlogrlim 24780

Description: The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)

**Assertion**
Ref	Expression
divlogrlim	⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0

Proof of Theorem divlogrlim Dummy variables 𝑐 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elioore 12493	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2		eliooord 12521	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
3	2	simpld 490	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 1 < 𝑥)
4	1, 3	rplogcld 24774	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → (log‘𝑥) ∈ ℝ⁺)
5	4	rprecred 12167	. . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℝ)
6	5	recnd 10385	. . . . . 6 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℂ)
7	6	rgen 3131	. . . . 5 ⊢ ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ
8	7	a1i 11	. . . 4 ⊢ (⊤ → ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ)
9		ioossre 12523	. . . . 5 ⊢ (1(,)+∞) ⊆ ℝ
10	9	a1i 11	. . . 4 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ)
11	8, 10	rlim0lt 14617	. . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0 ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)))
12	11	mptru 1664	. 2 ⊢ ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0 ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
13		id 22	. . . . 5 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ⁺)
14	13	rprecred 12167	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ)
15	14	reefcld 15190	. . 3 ⊢ (𝑦 ∈ ℝ⁺ → (exp‘(1 / 𝑦)) ∈ ℝ)
16	5	ad2antlr 718	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ)
17	1	ad2antlr 718	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ)
18	3	ad2antlr 718	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 < 𝑥)
19	17, 18	rplogcld 24774	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ⁺)
20	19	rpreccld 12166	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ⁺)
21	20	rpge0d 12160	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 0 ≤ (1 / (log‘𝑥)))
22	16, 21	absidd 14538	. . . . . 6 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) = (1 / (log‘𝑥)))
23		simpll 783	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑦 ∈ ℝ⁺)
24	4	ad2antlr 718	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ⁺)
25		simpr 479	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < 𝑥)
26		1rp 12116	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ⁺)
28	27	rpred 12156	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ)
29	28, 17, 18	ltled 10504	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ≤ 𝑥)
30	17, 27, 29	rpgecld 12195	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ⁺)
31	30	reeflogd 24769	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(log‘𝑥)) = 𝑥)
32	25, 31	breqtrrd 4901	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥)))
33	23	rprecred 12167	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) ∈ ℝ)
34	24	rpred 12156	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ)
35		eflt 15219	. . . . . . . . 9 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))))
36	33, 34, 35	syl2anc 579	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))))
37	32, 36	mpbird 249	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) < (log‘𝑥))
38	23, 24, 37	ltrec1d 12176	. . . . . 6 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) < 𝑦)
39	22, 38	eqbrtrd 4895	. . . . 5 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) < 𝑦)
40	39	ex 403	. . . 4 ⊢ ((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) → ((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
41	40	ralrimiva 3175	. . 3 ⊢ (𝑦 ∈ ℝ⁺ → ∀𝑥 ∈ (1(,)+∞)((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
42		breq1 4876	. . . 4 ⊢ (𝑐 = (exp‘(1 / 𝑦)) → (𝑐 < 𝑥 ↔ (exp‘(1 / 𝑦)) < 𝑥))
43	42	rspceaimv 3534	. . 3 ⊢ (((exp‘(1 / 𝑦)) ∈ ℝ ∧ ∀𝑥 ∈ (1(,)+∞)((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
44	15, 41, 43	syl2anc 579	. 2 ⊢ (𝑦 ∈ ℝ⁺ → ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
45	12, 44	mprgbir 3136	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ⊤wtru 1657 ∈ wcel 2164 ∀wral 3117 ∃wrex 3118 ⊆ wss 3798 class class class wbr 4873 ↦ cmpt 4952 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 1c1 10253 +∞cpnf 10388 < clt 10391 / cdiv 11009 ℝ⁺crp 12112 (,)cioo 12463 abscabs 14351 ⇝_𝑟 crli 14593 expce 15164 logclog 24700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ioc 12468 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155 df-fac 13354 df-bc 13383 df-hash 13411 df-shft 14184 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-limsup 14579 df-clim 14596 df-rlim 14597 df-sum 14794 df-ef 15170 df-sin 15172 df-cos 15173 df-pi 15175 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-hom 16329 df-cco 16330 df-rest 16436 df-topn 16437 df-0g 16455 df-gsum 16456 df-topgen 16457 df-pt 16458 df-prds 16461 df-xrs 16515 df-qtop 16520 df-imas 16521 df-xps 16523 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-mulg 17895 df-cntz 18100 df-cmn 18548 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-fbas 20103 df-fg 20104 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nei 21273 df-lp 21311 df-perf 21312 df-cn 21402 df-cnp 21403 df-haus 21490 df-tx 21736 df-hmeo 21929 df-fil 22020 df-fm 22112 df-flim 22113 df-flf 22114 df-xms 22495 df-ms 22496 df-tms 22497 df-cncf 23051 df-limc 24029 df-dv 24030 df-log 24702

This theorem is referenced by: logno1 24781 vmalogdivsum2 25640 2vmadivsumlem 25642 selberg4lem1 25662 pntrlog2bndlem2 25680 pntrlog2bndlem4 25682 pntrlog2bndlem5 25683

W3C validator