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Mirrors > Home > MPE Home > Th. List > logdivlt | Structured version Visualization version GIF version |
Description: The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 14-Mar-2014.) |
Ref | Expression |
---|---|
logdivlt | ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logdivlti 24366 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐴 < 𝐵) → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)) | |
2 | 1 | ex 450 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))) |
3 | 2 | 3expa 1265 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))) |
4 | 3 | an32s 846 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))) |
5 | 4 | adantrr 753 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))) |
6 | fveq2 6191 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → (log‘𝐴) = (log‘𝐵)) | |
7 | id 22 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
8 | 6, 7 | oveq12d 6668 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → ((log‘𝐴) / 𝐴) = ((log‘𝐵) / 𝐵)) |
9 | 8 | eqcomd 2628 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴)) |
10 | 9 | a1i 11 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 = 𝐵 → ((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴))) |
11 | logdivlti 24366 | . . . . . . . . . 10 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐵 < 𝐴) → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)) | |
12 | 11 | ex 450 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))) |
13 | 12 | 3expa 1265 | . . . . . . . 8 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))) |
14 | 13 | an32s 846 | . . . . . . 7 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))) |
15 | 14 | adantrr 753 | . . . . . 6 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ (𝐴 ∈ ℝ ∧ e ≤ 𝐴)) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))) |
16 | 15 | ancoms 469 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))) |
17 | 10, 16 | orim12d 883 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → (((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴) ∨ ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))) |
18 | 17 | con3d 148 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (¬ (((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴) ∨ ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)) → ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) |
19 | simpl 473 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ) | |
20 | epos 14935 | . . . . . . . 8 ⊢ 0 < e | |
21 | 0re 10040 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
22 | ere 14819 | . . . . . . . . 9 ⊢ e ∈ ℝ | |
23 | ltletr 10129 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵)) | |
24 | 21, 22, 23 | mp3an12 1414 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵)) |
25 | 20, 24 | mpani 712 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → (e ≤ 𝐵 → 0 < 𝐵)) |
26 | 25 | imp 445 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 0 < 𝐵) |
27 | 19, 26 | elrpd 11869 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ+) |
28 | relogcl 24322 | . . . . . 6 ⊢ (𝐵 ∈ ℝ+ → (log‘𝐵) ∈ ℝ) | |
29 | rerpdivcl 11861 | . . . . . 6 ⊢ (((log‘𝐵) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((log‘𝐵) / 𝐵) ∈ ℝ) | |
30 | 28, 29 | mpancom 703 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → ((log‘𝐵) / 𝐵) ∈ ℝ) |
31 | 27, 30 | syl 17 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → ((log‘𝐵) / 𝐵) ∈ ℝ) |
32 | simpl 473 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ) | |
33 | ltletr 10129 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴)) | |
34 | 21, 22, 33 | mp3an12 1414 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴)) |
35 | 20, 34 | mpani 712 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (e ≤ 𝐴 → 0 < 𝐴)) |
36 | 35 | imp 445 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 0 < 𝐴) |
37 | 32, 36 | elrpd 11869 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ+) |
38 | relogcl 24322 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
39 | rerpdivcl 11861 | . . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ+) → ((log‘𝐴) / 𝐴) ∈ ℝ) | |
40 | 38, 39 | mpancom 703 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) / 𝐴) ∈ ℝ) |
41 | 37, 40 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → ((log‘𝐴) / 𝐴) ∈ ℝ) |
42 | axlttri 10109 | . . . 4 ⊢ ((((log‘𝐵) / 𝐵) ∈ ℝ ∧ ((log‘𝐴) / 𝐴) ∈ ℝ) → (((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴) ↔ ¬ (((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴) ∨ ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))) | |
43 | 31, 41, 42 | syl2anr 495 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴) ↔ ¬ (((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴) ∨ ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))) |
44 | axlttri 10109 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) | |
45 | 44 | ad2ant2r 783 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) |
46 | 18, 43, 45 | 3imtr4d 283 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴) → 𝐴 < 𝐵)) |
47 | 5, 46 | impbid 202 | 1 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 < clt 10074 ≤ cle 10075 / cdiv 10684 ℝ+crp 11832 eceu 14793 logclog 24301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 |
This theorem is referenced by: logdivle 24368 bposlem7 25015 chebbnd1lem2 25159 chebbnd1lem3 25160 pntpbnd1a 25274 |
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