Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12076 | . . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ) | |
2 | eliooord 12104 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞)) | |
3 | 2 | simpld 474 | . . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 1 < 𝑥) |
4 | 1, 3 | rplogcld 24179 | . . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → (log‘𝑥) ∈ ℝ+) |
5 | 4 | rprecred 11759 | . . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℝ) |
6 | 5 | recnd 9947 | . . . . . 6 ⊢ (𝑥 ∈ (1(,)+∞) → (1 / (log‘𝑥)) ∈ ℂ) |
7 | 6 | rgen 2906 | . . . . 5 ⊢ ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → ∀𝑥 ∈ (1(,)+∞)(1 / (log‘𝑥)) ∈ ℂ) |
9 | ioossre 12106 | . . . . 5 ⊢ (1(,)+∞) ⊆ ℝ | |
10 | 9 | a1i 11 | . . . 4 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ) |
11 | 8, 10 | rlim0lt 14088 | . . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 ↔ ∀𝑦 ∈ ℝ+ ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))) |
12 | 11 | trud 1484 | . 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 11759 | . . . 4 ⊢ (𝑦 ∈ ℝ+ → (1 / 𝑦) ∈ ℝ) |
15 | 14 | reefcld 14657 | . . 3 ⊢ (𝑦 ∈ ℝ+ → (exp‘(1 / 𝑦)) ∈ ℝ) |
16 | 5 | ad2antlr 759 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ) |
17 | 1 | ad2antlr 759 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ) |
18 | 3 | ad2antlr 759 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 < 𝑥) |
19 | 17, 18 | rplogcld 24179 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ+) |
20 | 19 | rpreccld 11758 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) ∈ ℝ+) |
21 | 20 | rpge0d 11752 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 0 ≤ (1 / (log‘𝑥))) |
22 | 16, 21 | absidd 14009 | . . . . . 6 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) = (1 / (log‘𝑥))) |
23 | simpll 786 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑦 ∈ ℝ+) | |
24 | 4 | ad2antlr 759 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ+) |
25 | simpr 476 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < 𝑥) | |
26 | 1rp 11712 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ+ | |
27 | 26 | a1i 11 | . . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ+) |
28 | 27 | rpred 11748 | . . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ∈ ℝ) |
29 | 28, 17, 18 | ltled 10064 | . . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 1 ≤ 𝑥) |
30 | 17, 27, 29 | rpgecld 11787 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → 𝑥 ∈ ℝ+) |
31 | 30 | reeflogd 24174 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(log‘𝑥)) = 𝑥) |
32 | 25, 31 | breqtrrd 4611 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥))) |
33 | 23 | rprecred 11759 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) ∈ ℝ) |
34 | 24 | rpred 11748 | . . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (log‘𝑥) ∈ ℝ) |
35 | eflt 14686 | . . . . . . . . 9 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥)))) | |
36 | 33, 34, 35 | syl2anc 691 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → ((1 / 𝑦) < (log‘𝑥) ↔ (exp‘(1 / 𝑦)) < (exp‘(log‘𝑥)))) |
37 | 32, 36 | mpbird 246 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / 𝑦) < (log‘𝑥)) |
38 | 23, 24, 37 | ltrec1d 11768 | . . . . . 6 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (1 / (log‘𝑥)) < 𝑦) |
39 | 22, 38 | eqbrtrd 4605 | . . . . 5 ⊢ (((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (exp‘(1 / 𝑦)) < 𝑥) → (abs‘(1 / (log‘𝑥))) < 𝑦) |
40 | 39 | ex 449 | . . . 4 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑥 ∈ (1(,)+∞)) → ((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)) |
41 | 40 | ralrimiva 2949 | . . 3 ⊢ (𝑦 ∈ ℝ+ → ∀𝑥 ∈ (1(,)+∞)((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)) |
42 | breq1 4586 | . . . . . 6 ⊢ (𝑐 = (exp‘(1 / 𝑦)) → (𝑐 < 𝑥 ↔ (exp‘(1 / 𝑦)) < 𝑥)) | |
43 | 42 | imbi1d 330 | . . . . 5 ⊢ (𝑐 = (exp‘(1 / 𝑦)) → ((𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦) ↔ ((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))) |
44 | 43 | ralbidv 2969 | . . . 4 ⊢ (𝑐 = (exp‘(1 / 𝑦)) → (∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦) ↔ ∀𝑥 ∈ (1(,)+∞)((exp‘(1 / 𝑦)) < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦))) |
45 | 44 | rspcev 3282 | . . 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)) |
46 | 15, 41, 45 | syl2anc 691 | . 2 ⊢ (𝑦 ∈ ℝ+ → ∃𝑐 ∈ ℝ ∀𝑥 ∈ (1(,)+∞)(𝑐 < 𝑥 → (abs‘(1 / (log‘𝑥))) < 𝑦)) |
47 | 12, 46 | mprgbir 2911 | 1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 +∞cpnf 9950 < clt 9953 / cdiv 10563 ℝ+crp 11708 (,)cioo 12046 abscabs 13822 ⇝𝑟 crli 14064 expce 14631 logclog 24105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 |
This theorem is referenced by: logno1 24182 vmalogdivsum2 25027 2vmadivsumlem 25029 selberg4lem1 25049 pntrlog2bndlem2 25067 pntrlog2bndlem4 25069 pntrlog2bndlem5 25070 |
Copyright terms: Public domain | W3C validator |