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Mirrors > Home > MPE Home > Th. List > logdiflbnd | Structured version Visualization version GIF version |
Description: Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.) |
Ref | Expression |
---|---|
logdiflbnd | ⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 12737 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpge0 12742 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) | |
3 | 1, 2 | ge0p1rpd 12801 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 1) ∈ ℝ+) |
4 | 3 | rprecred 12782 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ∈ ℝ) |
5 | 1red 10977 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℝ) | |
6 | 0le1 11498 | . . . . . . 7 ⊢ 0 ≤ 1 | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 1) |
8 | 5, 3, 7 | divge0d 12811 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ (1 / (𝐴 + 1))) |
9 | id 22 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+) | |
10 | 5, 9 | ltaddrp2d 12805 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 1 < (𝐴 + 1)) |
11 | 1, 5 | readdcld 11005 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 1) ∈ ℝ) |
12 | 11 | recnd 11004 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 1) ∈ ℂ) |
13 | 12 | mulid1d 10993 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → ((𝐴 + 1) · 1) = (𝐴 + 1)) |
14 | 10, 13 | breqtrrd 5107 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 1 < ((𝐴 + 1) · 1)) |
15 | 5, 5, 3 | ltdivmuld 12822 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → ((1 / (𝐴 + 1)) < 1 ↔ 1 < ((𝐴 + 1) · 1))) |
16 | 14, 15 | mpbird 256 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) < 1) |
17 | 4, 8, 16 | eflegeo 15828 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (exp‘(1 / (𝐴 + 1))) ≤ (1 / (1 − (1 / (𝐴 + 1))))) |
18 | 5 | recnd 11004 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℂ) |
19 | 3 | rpne0d 12776 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 1) ≠ 0) |
20 | 12, 18, 12, 19 | divsubdird 11790 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1)))) |
21 | 1 | recnd 11004 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) |
22 | 21, 18 | pncand 11333 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → ((𝐴 + 1) − 1) = 𝐴) |
23 | 22 | oveq1d 7286 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (𝐴 / (𝐴 + 1))) |
24 | 12, 19 | dividd 11749 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → ((𝐴 + 1) / (𝐴 + 1)) = 1) |
25 | 24 | oveq1d 7286 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1))) = (1 − (1 / (𝐴 + 1)))) |
26 | 20, 23, 25 | 3eqtr3rd 2789 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (1 − (1 / (𝐴 + 1))) = (𝐴 / (𝐴 + 1))) |
27 | 26 | oveq2d 7287 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / (1 − (1 / (𝐴 + 1)))) = (1 / (𝐴 / (𝐴 + 1)))) |
28 | rpne0 12745 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
29 | 21, 12, 28, 19 | recdivd 11768 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 / (𝐴 + 1))) = ((𝐴 + 1) / 𝐴)) |
30 | 27, 29 | eqtrd 2780 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (1 / (1 − (1 / (𝐴 + 1)))) = ((𝐴 + 1) / 𝐴)) |
31 | 17, 30 | breqtrd 5105 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴)) |
32 | 4 | rpefcld 15812 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (exp‘(1 / (𝐴 + 1))) ∈ ℝ+) |
33 | 3, 9 | rpdivcld 12788 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((𝐴 + 1) / 𝐴) ∈ ℝ+) |
34 | 32, 33 | logled 25780 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴) ↔ (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴)))) |
35 | 31, 34 | mpbid 231 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴))) |
36 | 4 | relogefd 25781 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘(1 / (𝐴 + 1)))) = (1 / (𝐴 + 1))) |
37 | 3, 9 | relogdivd 25779 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘((𝐴 + 1) / 𝐴)) = ((log‘(𝐴 + 1)) − (log‘𝐴))) |
38 | 35, 36, 37 | 3brtr3d 5110 | 1 ⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 (class class class)co 7271 0cc0 10872 1c1 10873 + caddc 10875 · cmul 10877 < clt 11010 ≤ cle 11011 − cmin 11205 / cdiv 11632 ℝ+crp 12729 expce 15769 logclog 25708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ioc 13083 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-fac 13986 df-bc 14015 df-hash 14043 df-shft 14776 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-limsup 15178 df-clim 15195 df-rlim 15196 df-sum 15396 df-ef 15775 df-sin 15777 df-cos 15778 df-pi 15780 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cncf 24039 df-limc 25028 df-dv 25029 df-log 25710 |
This theorem is referenced by: lgamgulmlem3 26178 |
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