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Theorem divlogrlim 26657

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 13400	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2		eliooord 13429	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
3	2	simpld 493	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 1 < 𝑥)
4	1, 3	rplogcld 26651	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → (log‘𝑥) ∈ ℝ⁺)
5	4	rprecred 13073	. . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℝ)
6	5	recnd 11281	. . . . . 6 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℂ)
7	6	rgen 3053	. . . . 5 ⊢ ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ
8	7	a1i 11	. . . 4 ⊢ (⊤ → ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ)
9		ioossre 13431	. . . . 5 ⊢ (1(,)+∞) ⊆ ℝ
10	9	a1i 11	. . . 4 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ)
11	8, 10	rlim0lt 15504	. . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0 ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)))
12	11	mptru 1541	. 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 13073	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ)
15	14	reefcld 16083	. . 3 ⊢ (𝑦 ∈ ℝ⁺ → (exp‘(1 / 𝑦)) ∈ ℝ)
16	5	ad2antlr 725	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ)
17	1	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ)
18	3	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 < 𝑥)
19	17, 18	rplogcld 26651	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ⁺)
20	19	rpreccld 13072	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ⁺)
21	20	rpge0d 13066	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 0 ≤ (1 / (log‘𝑥)))
22	16, 21	absidd 15420	. . . . . 6 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) = (1 / (log‘𝑥)))
23		simpll 765	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑦 ∈ ℝ⁺)
24	4	ad2antlr 725	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ⁺)
25		simpr 483	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < 𝑥)
26		1rp 13024	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ⁺)
28	27	rpred 13062	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ)
29	28, 17, 18	ltled 11401	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ≤ 𝑥)
30	17, 27, 29	rpgecld 13101	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ⁺)
31	30	reeflogd 26646	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(log‘𝑥)) = 𝑥)
32	25, 31	breqtrrd 5172	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥)))
33	23	rprecred 13073	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) ∈ ℝ)
34	24	rpred 13062	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ)
35		eflt 16112	. . . . . . . . 9 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))))
36	33, 34, 35	syl2anc 582	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))))
37	32, 36	mpbird 256	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) < (log‘𝑥))
38	23, 24, 37	ltrec1d 13082	. . . . . 6 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) < 𝑦)
39	22, 38	eqbrtrd 5166	. . . . 5 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) < 𝑦)
40	39	ex 411	. . . 4 ⊢ ((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) → ((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
41	40	ralrimiva 3136	. . 3 ⊢ (𝑦 ∈ ℝ⁺ → ∀𝑥 ∈ (1(,)+∞)((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
42		breq1 5147	. . . 4 ⊢ (𝑐 = (exp‘(1 / 𝑦)) → (𝑐 < 𝑥 ↔ (exp‘(1 / 𝑦)) < 𝑥))
43	42	rspceaimv 3614	. . 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 582	. 2 ⊢ (𝑦 ∈ ℝ⁺ → ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
45	12, 44	mprgbir 3058	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ⊤wtru 1535 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 ⊆ wss 3947 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6544 (class class class)co 7414 ℂcc 11145 ℝcr 11146 0cc0 11147 1c1 11148 +∞cpnf 11284 < clt 11287 / cdiv 11910 ℝ⁺crp 13020 (,)cioo 13370 abscabs 15232 ⇝_𝑟 crli 15480 expce 16056 logclog 26576

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-inf2 9675 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 ax-addf 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9397 df-fi 9445 df-sup 9476 df-inf 9477 df-oi 9544 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13374 df-ioc 13375 df-ico 13376 df-icc 13377 df-fz 13531 df-fzo 13674 df-fl 13804 df-mod 13882 df-seq 14014 df-exp 14074 df-fac 14284 df-bc 14313 df-hash 14341 df-shft 15065 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-limsup 15466 df-clim 15483 df-rlim 15484 df-sum 15684 df-ef 16062 df-sin 16064 df-cos 16065 df-pi 16067 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-starv 17274 df-sca 17275 df-vsca 17276 df-ip 17277 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-hom 17283 df-cco 17284 df-rest 17430 df-topn 17431 df-0g 17449 df-gsum 17450 df-topgen 17451 df-pt 17452 df-prds 17455 df-xrs 17510 df-qtop 17515 df-imas 17516 df-xps 17518 df-mre 17592 df-mrc 17593 df-acs 17595 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19774 df-psmet 21329 df-xmet 21330 df-met 21331 df-bl 21332 df-mopn 21333 df-fbas 21334 df-fg 21335 df-cnfld 21338 df-top 22882 df-topon 22899 df-topsp 22921 df-bases 22935 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24312 df-ms 24313 df-tms 24314 df-cncf 24884 df-limc 25881 df-dv 25882 df-log 26578

This theorem is referenced by: logno1 26658 vmalogdivsum2 27562 2vmadivsumlem 27564 selberg4lem1 27584 pntrlog2bndlem2 27602 pntrlog2bndlem4 27604 pntrlog2bndlem5 27605

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