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Theorem divlogrlim 25218

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 12769	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2		eliooord 12797	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
3	2	simpld 497	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 1 < 𝑥)
4	1, 3	rplogcld 25212	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → (log‘𝑥) ∈ ℝ⁺)
5	4	rprecred 12443	. . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℝ)
6	5	recnd 10669	. . . . . 6 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℂ)
7	6	rgen 3148	. . . . 5 ⊢ ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ
8	7	a1i 11	. . . 4 ⊢ (⊤ → ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ)
9		ioossre 12799	. . . . 5 ⊢ (1(,)+∞) ⊆ ℝ
10	9	a1i 11	. . . 4 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ)
11	8, 10	rlim0lt 14866	. . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0 ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)))
12	11	mptru 1544	. 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 12443	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ)
15	14	reefcld 15441	. . 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 25212	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ⁺)
20	19	rpreccld 12442	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ⁺)
21	20	rpge0d 12436	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 0 ≤ (1 / (log‘𝑥)))
22	16, 21	absidd 14782	. . . . . 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 487	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < 𝑥)
26		1rp 12394	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ⁺)
28	27	rpred 12432	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ)
29	28, 17, 18	ltled 10788	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ≤ 𝑥)
30	17, 27, 29	rpgecld 12471	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ⁺)
31	30	reeflogd 25207	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(log‘𝑥)) = 𝑥)
32	25, 31	breqtrrd 5094	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥)))
33	23	rprecred 12443	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) ∈ ℝ)
34	24	rpred 12432	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ)
35		eflt 15470	. . . . . . . . 9 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))))
36	33, 34, 35	syl2anc 586	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))))
37	32, 36	mpbird 259	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) < (log‘𝑥))
38	23, 24, 37	ltrec1d 12452	. . . . . 6 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) < 𝑦)
39	22, 38	eqbrtrd 5088	. . . . 5 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) < 𝑦)
40	39	ex 415	. . . 4 ⊢ ((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) → ((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
41	40	ralrimiva 3182	. . 3 ⊢ (𝑦 ∈ ℝ⁺ → ∀𝑥 ∈ (1(,)+∞)((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
42		breq1 5069	. . . 4 ⊢ (𝑐 = (exp‘(1 / 𝑦)) → (𝑐 < 𝑥 ↔ (exp‘(1 / 𝑦)) < 𝑥))
43	42	rspceaimv 3628	. . 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 586	. 2 ⊢ (𝑦 ∈ ℝ⁺ → ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
45	12, 44	mprgbir 3153	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ⊤wtru 1538 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 +∞cpnf 10672 < clt 10675 / cdiv 11297 ℝ⁺crp 12390 (,)cioo 12739 abscabs 14593 ⇝_𝑟 crli 14842 expce 15415 logclog 25138

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-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-sin 15423 df-cos 15424 df-pi 15426 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-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-log 25140

This theorem is referenced by: logno1 25219 vmalogdivsum2 26114 2vmadivsumlem 26116 selberg4lem1 26136 pntrlog2bndlem2 26154 pntrlog2bndlem4 26156 pntrlog2bndlem5 26157

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