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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 13065 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpge0 13070 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) | |
3 | 1, 2 | ge0p1rpd 13129 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 1) ∈ ℝ+) |
4 | 3 | rprecred 13110 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ∈ ℝ) |
5 | 1red 11291 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℝ) | |
6 | 0le1 11813 | . . . . . . 7 ⊢ 0 ≤ 1 | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 1) |
8 | 5, 3, 7 | divge0d 13139 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ (1 / (𝐴 + 1))) |
9 | id 22 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+) | |
10 | 5, 9 | ltaddrp2d 13133 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 1 < (𝐴 + 1)) |
11 | 1, 5 | readdcld 11319 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 1) ∈ ℝ) |
12 | 11 | recnd 11318 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 1) ∈ ℂ) |
13 | 12 | mulridd 11307 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → ((𝐴 + 1) · 1) = (𝐴 + 1)) |
14 | 10, 13 | breqtrrd 5194 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 1 < ((𝐴 + 1) · 1)) |
15 | 5, 5, 3 | ltdivmuld 13150 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → ((1 / (𝐴 + 1)) < 1 ↔ 1 < ((𝐴 + 1) · 1))) |
16 | 14, 15 | mpbird 257 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) < 1) |
17 | 4, 8, 16 | eflegeo 16169 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (exp‘(1 / (𝐴 + 1))) ≤ (1 / (1 − (1 / (𝐴 + 1))))) |
18 | 5 | recnd 11318 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℂ) |
19 | 3 | rpne0d 13104 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 1) ≠ 0) |
20 | 12, 18, 12, 19 | divsubdird 12109 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1)))) |
21 | 1 | recnd 11318 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) |
22 | 21, 18 | pncand 11648 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → ((𝐴 + 1) − 1) = 𝐴) |
23 | 22 | oveq1d 7463 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (𝐴 / (𝐴 + 1))) |
24 | 12, 19 | dividd 12068 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → ((𝐴 + 1) / (𝐴 + 1)) = 1) |
25 | 24 | oveq1d 7463 | . . . . . . 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 7464 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / (1 − (1 / (𝐴 + 1)))) = (1 / (𝐴 / (𝐴 + 1)))) |
28 | rpne0 13073 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
29 | 21, 12, 28, 19 | recdivd 12087 | . . . . 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 5192 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴)) |
32 | 4 | rpefcld 16153 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (exp‘(1 / (𝐴 + 1))) ∈ ℝ+) |
33 | 3, 9 | rpdivcld 13116 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((𝐴 + 1) / 𝐴) ∈ ℝ+) |
34 | 32, 33 | logled 26687 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴) ↔ (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴)))) |
35 | 31, 34 | mpbid 232 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴))) |
36 | 4 | relogefd 26688 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘(1 / (𝐴 + 1)))) = (1 / (𝐴 + 1))) |
37 | 3, 9 | relogdivd 26686 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘((𝐴 + 1) / 𝐴)) = ((log‘(𝐴 + 1)) − (log‘𝐴))) |
38 | 35, 36, 37 | 3brtr3d 5197 | 1 ⊢ (𝐴 ∈ ℝ+ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℝ+crp 13057 expce 16109 logclog 26614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 |
This theorem is referenced by: lgamgulmlem3 27092 |
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