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Theorem divlogrlim 26599

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 13328	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2		eliooord 13358	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
3	2	simpld 494	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 1 < 𝑥)
4	1, 3	rplogcld 26593	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → (log‘𝑥) ∈ ℝ⁺)
5	4	rprecred 12997	. . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℝ)
6	5	recnd 11173	. . . . . 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 13360	. . . . 5 ⊢ (1(,)+∞) ⊆ ℝ
10	9	a1i 11	. . . 4 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ)
11	8, 10	rlim0lt 15471	. . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0 ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)))
12	11	mptru 1549	. 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 12997	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ)
15	14	reefcld 16053	. . 3 ⊢ (𝑦 ∈ ℝ⁺ → (exp‘(1 / 𝑦)) ∈ ℝ)
16	5	ad2antlr 728	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ)
17	1	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ)
18	3	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 < 𝑥)
19	17, 18	rplogcld 26593	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ⁺)
20	19	rpreccld 12996	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ⁺)
21	20	rpge0d 12990	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 0 ≤ (1 / (log‘𝑥)))
22	16, 21	absidd 15385	. . . . . 6 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) = (1 / (log‘𝑥)))
23		simpll 767	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑦 ∈ ℝ⁺)
24	4	ad2antlr 728	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ⁺)
25		simpr 484	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < 𝑥)
26		1rp 12946	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ⁺)
28	27	rpred 12986	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ)
29	28, 17, 18	ltled 11294	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ≤ 𝑥)
30	17, 27, 29	rpgecld 13025	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ⁺)
31	30	reeflogd 26588	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(log‘𝑥)) = 𝑥)
32	25, 31	breqtrrd 5113	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥)))
33	23	rprecred 12997	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) ∈ ℝ)
34	24	rpred 12986	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ)
35		eflt 16084	. . . . . . . . 9 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))))
36	33, 34, 35	syl2anc 585	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))))
37	32, 36	mpbird 257	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) < (log‘𝑥))
38	23, 24, 37	ltrec1d 13006	. . . . . 6 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) < 𝑦)
39	22, 38	eqbrtrd 5107	. . . . 5 ⊢ (((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) < 𝑦)
40	39	ex 412	. . . 4 ⊢ ((𝑦 ∈ ℝ⁺ ∧ 𝑥 ∈ (1(,)+∞)) → ((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
41	40	ralrimiva 3129	. . 3 ⊢ (𝑦 ∈ ℝ⁺ → ∀𝑥 ∈ (1(,)+∞)((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))
42		breq1 5088	. . . 4 ⊢ (𝑐 = (exp‘(1 / 𝑦)) → (𝑐 < 𝑥 ↔ (exp‘(1 / 𝑦)) < 𝑥))
43	42	rspceaimv 3570	. . 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 585	. 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 206 ∧ wa 395 ⊤wtru 1543 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 ⊆ wss 3889 class class class wbr 5085 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 +∞cpnf 11176 < clt 11179 / cdiv 11807 ℝ⁺crp 12942 (,)cioo 13298 abscabs 15196 ⇝_𝑟 crli 15447 expce 16026 logclog 26518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520

This theorem is referenced by: logno1 26600 vmalogdivsum2 27501 2vmadivsumlem 27503 selberg4lem1 27523 pntrlog2bndlem2 27541 pntrlog2bndlem4 27543 pntrlog2bndlem5 27544

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