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Mirrors > Home > MPE Home > Th. List > Mathboxes > rege1logbrege0 | Structured version Visualization version GIF version |
Description: The general logarithm, with a real base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.) |
Ref | Expression |
---|---|
rege1logbrege0 | ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 logb 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10378 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
2 | elicopnf 12587 | . . . . . . 7 ⊢ (1 ∈ ℝ → (𝑋 ∈ (1[,)+∞) ↔ (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋))) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ (𝑋 ∈ (1[,)+∞) ↔ (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋)) |
4 | id 22 | . . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋)) | |
5 | 3, 4 | sylbi 209 | . . . . 5 ⊢ (𝑋 ∈ (1[,)+∞) → (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋)) |
6 | 5 | adantl 475 | . . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋)) |
7 | logge0 24799 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 0 ≤ (log‘𝑋)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (log‘𝑋)) |
9 | simpl 476 | . . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℝ) | |
10 | 0lt1 10900 | . . . . . . . . . 10 ⊢ 0 < 1 | |
11 | 0red 10382 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 0 ∈ ℝ) | |
12 | 1red 10379 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 1 ∈ ℝ) | |
13 | id 22 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℝ) | |
14 | ltletr 10470 | . . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑋) → 0 < 𝑋)) | |
15 | 11, 12, 13, 14 | syl3anc 1439 | . . . . . . . . . 10 ⊢ (𝑋 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝑋) → 0 < 𝑋)) |
16 | 10, 15 | mpani 686 | . . . . . . . . 9 ⊢ (𝑋 ∈ ℝ → (1 ≤ 𝑋 → 0 < 𝑋)) |
17 | 16 | imp 397 | . . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 0 < 𝑋) |
18 | 9, 17 | elrpd 12183 | . . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℝ+) |
19 | 3, 18 | sylbi 209 | . . . . . 6 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ ℝ+) |
20 | 19 | relogcld 24817 | . . . . 5 ⊢ (𝑋 ∈ (1[,)+∞) → (log‘𝑋) ∈ ℝ) |
21 | 20 | adantl 475 | . . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝑋) ∈ ℝ) |
22 | 1xr 10438 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
23 | elioopnf 12585 | . . . . . . . 8 ⊢ (1 ∈ ℝ* → (𝐵 ∈ (1(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 1 < 𝐵))) | |
24 | 22, 23 | ax-mp 5 | . . . . . . 7 ⊢ (𝐵 ∈ (1(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 1 < 𝐵)) |
25 | simpl 476 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℝ) | |
26 | 0red 10382 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ) | |
27 | 1red 10379 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 1 ∈ ℝ) | |
28 | id 22 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ) | |
29 | lttr 10455 | . . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝐵) → 0 < 𝐵)) | |
30 | 26, 27, 28, 29 | syl3anc 1439 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → ((0 < 1 ∧ 1 < 𝐵) → 0 < 𝐵)) |
31 | 10, 30 | mpani 686 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (1 < 𝐵 → 0 < 𝐵)) |
32 | 31 | imp 397 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 0 < 𝐵) |
33 | 25, 32 | elrpd 12183 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℝ+) |
34 | 24, 33 | sylbi 209 | . . . . . 6 ⊢ (𝐵 ∈ (1(,)+∞) → 𝐵 ∈ ℝ+) |
35 | 34 | relogcld 24817 | . . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → (log‘𝐵) ∈ ℝ) |
36 | 35 | adantr 474 | . . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝐵) ∈ ℝ) |
37 | regt1loggt0 43359 | . . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → 0 < (log‘𝐵)) | |
38 | 37 | adantr 474 | . . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 < (log‘𝐵)) |
39 | ge0div 11247 | . . . 4 ⊢ (((log‘𝑋) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ ∧ 0 < (log‘𝐵)) → (0 ≤ (log‘𝑋) ↔ 0 ≤ ((log‘𝑋) / (log‘𝐵)))) | |
40 | 21, 36, 38, 39 | syl3anc 1439 | . . 3 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (0 ≤ (log‘𝑋) ↔ 0 ≤ ((log‘𝑋) / (log‘𝐵)))) |
41 | 8, 40 | mpbid 224 | . 2 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ ((log‘𝑋) / (log‘𝐵))) |
42 | recn 10364 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
43 | 42 | adantr 474 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℂ) |
44 | 32 | gt0ne0d 10942 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ≠ 0) |
45 | 27, 28 | ltlend 10523 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → (1 < 𝐵 ↔ (1 ≤ 𝐵 ∧ 𝐵 ≠ 1))) |
46 | 45 | simplbda 495 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ≠ 1) |
47 | 43, 44, 46 | 3jca 1119 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) |
48 | eldifpr 4426 | . . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) | |
49 | 47, 24, 48 | 3imtr4i 284 | . . 3 ⊢ (𝐵 ∈ (1(,)+∞) → 𝐵 ∈ (ℂ ∖ {0, 1})) |
50 | recn 10364 | . . . . . 6 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℂ) | |
51 | 50 | adantr 474 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℂ) |
52 | 17 | gt0ne0d 10942 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ≠ 0) |
53 | 51, 52 | jca 507 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0)) |
54 | eldifsn 4550 | . . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0)) | |
55 | 53, 3, 54 | 3imtr4i 284 | . . 3 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ (ℂ ∖ {0})) |
56 | logbval 24955 | . . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) | |
57 | 49, 55, 56 | syl2an 589 | . 2 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) |
58 | 41, 57 | breqtrrd 4916 | 1 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 logb 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 {csn 4398 {cpr 4400 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 +∞cpnf 10410 ℝ*cxr 10412 < clt 10413 ≤ cle 10414 / cdiv 11035 ℝ+crp 12142 (,)cioo 12492 [,)cico 12494 logclog 24749 logb clogb 24953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ioc 12497 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-fl 12917 df-mod 12993 df-seq 13125 df-exp 13184 df-fac 13385 df-bc 13414 df-hash 13442 df-shft 14220 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-limsup 14619 df-clim 14636 df-rlim 14637 df-sum 14834 df-ef 15209 df-sin 15211 df-cos 15212 df-pi 15214 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-pt 16502 df-prds 16505 df-xrs 16559 df-qtop 16564 df-imas 16565 df-xps 16567 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-mulg 17939 df-cntz 18144 df-cmn 18592 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-fbas 20150 df-fg 20151 df-cnfld 20154 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cld 21242 df-ntr 21243 df-cls 21244 df-nei 21321 df-lp 21359 df-perf 21360 df-cn 21450 df-cnp 21451 df-haus 21538 df-tx 21785 df-hmeo 21978 df-fil 22069 df-fm 22161 df-flim 22162 df-flf 22163 df-xms 22544 df-ms 22545 df-tms 22546 df-cncf 23100 df-limc 24078 df-dv 24079 df-log 24751 df-logb 24954 |
This theorem is referenced by: rege1logbzge0 43382 |
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