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Mirrors > Home > MPE Home > Th. List > divlogrlim | Structured version Visualization version GIF version |
Description: The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
divlogrlim | ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 12453 | . . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ) | |
2 | eliooord 12481 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞)) | |
3 | 2 | simpld 489 | . . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 1 < 𝑥) |
4 | 1, 3 | rplogcld 24715 | . . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → (log‘𝑥) ∈ ℝ+) |
5 | 4 | rprecred 12127 | . . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℝ) |
6 | 5 | recnd 10358 | . . . . . 6 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℂ) |
7 | 6 | rgen 3104 | . . . . 5 ⊢ ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ) |
9 | ioossre 12483 | . . . . 5 ⊢ (1(,)+∞) ⊆ ℝ | |
10 | 9 | a1i 11 | . . . 4 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ) |
11 | 8, 10 | rlim0lt 14580 | . . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 ↔ ∀𝑦 ∈ ℝ+ ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))) |
12 | 11 | mptru 1661 | . 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 12127 | . . . 4 ⊢ (𝑦 ∈ ℝ+ → (1 / 𝑦) ∈ ℝ) |
15 | 14 | reefcld 15153 | . . 3 ⊢ (𝑦 ∈ ℝ+ → (exp‘(1 / 𝑦)) ∈ ℝ) |
16 | 5 | ad2antlr 719 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ) |
17 | 1 | ad2antlr 719 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ) |
18 | 3 | ad2antlr 719 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 < 𝑥) |
19 | 17, 18 | rplogcld 24715 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ+) |
20 | 19 | rpreccld 12126 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ+) |
21 | 20 | rpge0d 12120 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 0 ≤ (1 / (log‘𝑥))) |
22 | 16, 21 | absidd 14501 | . . . . . 6 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) = (1 / (log‘𝑥))) |
23 | simpll 784 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑦 ∈ ℝ+) | |
24 | 4 | ad2antlr 719 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ+) |
25 | simpr 478 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < 𝑥) | |
26 | 1rp 12077 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ+ | |
27 | 26 | a1i 11 | . . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ+) |
28 | 27 | rpred 12116 | . . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ) |
29 | 28, 17, 18 | ltled 10476 | . . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ≤ 𝑥) |
30 | 17, 27, 29 | rpgecld 12155 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ+) |
31 | 30 | reeflogd 24710 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(log‘𝑥)) = 𝑥) |
32 | 25, 31 | breqtrrd 4872 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))) |
33 | 23 | rprecred 12127 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) ∈ ℝ) |
34 | 24 | rpred 12116 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ) |
35 | eflt 15182 | . . . . . . . . 9 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥)))) | |
36 | 33, 34, 35 | syl2anc 580 | . . . . . . . 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 12136 | . . . . . 6 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) < 𝑦) |
39 | 22, 38 | eqbrtrd 4866 | . . . . 5 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) < 𝑦) |
40 | 39 | ex 402 | . . . 4 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) → ((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)) |
41 | 40 | ralrimiva 3148 | . . 3 ⊢ (𝑦 ∈ ℝ+ → ∀𝑥 ∈ (1(,)+∞)((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)) |
42 | breq1 4847 | . . . 4 ⊢ (𝑐 = (exp‘(1 / 𝑦)) → (𝑐 < 𝑥 ↔ (exp‘(1 / 𝑦)) < 𝑥)) | |
43 | 42 | rspceaimv 3506 | . . 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 580 | . 2 ⊢ (𝑦 ∈ ℝ+ → ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)) |
45 | 12, 44 | mprgbir 3109 | 1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ⊤wtru 1654 ∈ wcel 2157 ∀wral 3090 ∃wrex 3091 ⊆ wss 3770 class class class wbr 4844 ↦ cmpt 4923 ‘cfv 6102 (class class class)co 6879 ℂcc 10223 ℝcr 10224 0cc0 10225 1c1 10226 +∞cpnf 10361 < clt 10364 / cdiv 10977 ℝ+crp 12073 (,)cioo 12423 abscabs 14314 ⇝𝑟 crli 14556 expce 15127 logclog 24641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-pre-sup 10303 ax-addf 10304 ax-mulf 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-of 7132 df-om 7301 df-1st 7402 df-2nd 7403 df-supp 7534 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-2o 7801 df-oadd 7804 df-er 7983 df-map 8098 df-pm 8099 df-ixp 8150 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-fsupp 8519 df-fi 8560 df-sup 8591 df-inf 8592 df-oi 8658 df-card 9052 df-cda 9279 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-9 11382 df-n0 11580 df-z 11666 df-dec 11783 df-uz 11930 df-q 12033 df-rp 12074 df-xneg 12192 df-xadd 12193 df-xmul 12194 df-ioo 12427 df-ioc 12428 df-ico 12429 df-icc 12430 df-fz 12580 df-fzo 12720 df-fl 12847 df-mod 12923 df-seq 13055 df-exp 13114 df-fac 13313 df-bc 13342 df-hash 13370 df-shft 14147 df-cj 14179 df-re 14180 df-im 14181 df-sqrt 14315 df-abs 14316 df-limsup 14542 df-clim 14559 df-rlim 14560 df-sum 14757 df-ef 15133 df-sin 15135 df-cos 15136 df-pi 15138 df-struct 16185 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-plusg 16279 df-mulr 16280 df-starv 16281 df-sca 16282 df-vsca 16283 df-ip 16284 df-tset 16285 df-ple 16286 df-ds 16288 df-unif 16289 df-hom 16290 df-cco 16291 df-rest 16397 df-topn 16398 df-0g 16416 df-gsum 16417 df-topgen 16418 df-pt 16419 df-prds 16422 df-xrs 16476 df-qtop 16481 df-imas 16482 df-xps 16484 df-mre 16560 df-mrc 16561 df-acs 16563 df-mgm 17556 df-sgrp 17598 df-mnd 17609 df-submnd 17650 df-mulg 17856 df-cntz 18061 df-cmn 18509 df-psmet 20059 df-xmet 20060 df-met 20061 df-bl 20062 df-mopn 20063 df-fbas 20064 df-fg 20065 df-cnfld 20068 df-top 21026 df-topon 21043 df-topsp 21065 df-bases 21078 df-cld 21151 df-ntr 21152 df-cls 21153 df-nei 21230 df-lp 21268 df-perf 21269 df-cn 21359 df-cnp 21360 df-haus 21447 df-tx 21693 df-hmeo 21886 df-fil 21977 df-fm 22069 df-flim 22070 df-flf 22071 df-xms 22452 df-ms 22453 df-tms 22454 df-cncf 23008 df-limc 23970 df-dv 23971 df-log 24643 |
This theorem is referenced by: logno1 24722 vmalogdivsum2 25578 2vmadivsumlem 25580 selberg4lem1 25600 pntrlog2bndlem2 25618 pntrlog2bndlem4 25620 pntrlog2bndlem5 25621 |
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