![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fllogbd | Structured version Visualization version GIF version |
Description: A real number is between the base of a logarithm to the power of the floor of the logarithm of the number and the base of the logarithm to the power of the floor of the logarithm of the number plus one. (Contributed by AV, 23-May-2020.) |
Ref | Expression |
---|---|
fllogbd.b | ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) |
fllogbd.x | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
fllogbd.e | ⊢ 𝐸 = (⌊‘(𝐵 logb 𝑋)) |
Ref | Expression |
---|---|
fllogbd | ⊢ (𝜑 → ((𝐵↑𝐸) ≤ 𝑋 ∧ 𝑋 < (𝐵↑(𝐸 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fllogbd.e | . . . . 5 ⊢ 𝐸 = (⌊‘(𝐵 logb 𝑋)) | |
2 | fllogbd.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) | |
3 | fllogbd.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
4 | relogbzcl 24952 | . . . . . . 7 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ) | |
5 | 2, 3, 4 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → (𝐵 logb 𝑋) ∈ ℝ) |
6 | flle 12919 | . . . . . 6 ⊢ ((𝐵 logb 𝑋) ∈ ℝ → (⌊‘(𝐵 logb 𝑋)) ≤ (𝐵 logb 𝑋)) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐵 logb 𝑋)) ≤ (𝐵 logb 𝑋)) |
8 | 1, 7 | syl5eqbr 4921 | . . . 4 ⊢ (𝜑 → 𝐸 ≤ (𝐵 logb 𝑋)) |
9 | eluzelz 12002 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℤ) | |
10 | 2, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
11 | 10 | zred 11834 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
12 | eluz2b1 12066 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘2) ↔ (𝐵 ∈ ℤ ∧ 1 < 𝐵)) | |
13 | 12 | simprbi 492 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) → 1 < 𝐵) |
14 | 2, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 1 < 𝐵) |
15 | 5 | flcld 12918 | . . . . . . 7 ⊢ (𝜑 → (⌊‘(𝐵 logb 𝑋)) ∈ ℤ) |
16 | 1, 15 | syl5eqel 2862 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℤ) |
17 | 16 | zred 11834 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
18 | 11, 14, 17, 5 | cxpled 24903 | . . . 4 ⊢ (𝜑 → (𝐸 ≤ (𝐵 logb 𝑋) ↔ (𝐵↑𝑐𝐸) ≤ (𝐵↑𝑐(𝐵 logb 𝑋)))) |
19 | 8, 18 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐𝐸) ≤ (𝐵↑𝑐(𝐵 logb 𝑋))) |
20 | 10 | zcnd 11835 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | eluz2nn 12032 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℕ) | |
22 | 2, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ) |
23 | 22 | nnne0d 11425 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 0) |
24 | 20, 23, 16 | cxpexpzd 24894 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐𝐸) = (𝐵↑𝐸)) |
25 | eluz2cnn0n1 43309 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ (ℂ ∖ {0, 1})) | |
26 | 2, 25 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ {0, 1})) |
27 | rpcnne0 12157 | . . . . . 6 ⊢ (𝑋 ∈ ℝ+ → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0)) | |
28 | eldifsn 4549 | . . . . . 6 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0)) | |
29 | 27, 28 | sylibr 226 | . . . . 5 ⊢ (𝑋 ∈ ℝ+ → 𝑋 ∈ (ℂ ∖ {0})) |
30 | 3, 29 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (ℂ ∖ {0})) |
31 | cxplogb 24964 | . . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) | |
32 | 26, 30, 31 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) |
33 | 19, 24, 32 | 3brtr3d 4917 | . 2 ⊢ (𝜑 → (𝐵↑𝐸) ≤ 𝑋) |
34 | flltp1 12920 | . . . . . 6 ⊢ ((𝐵 logb 𝑋) ∈ ℝ → (𝐵 logb 𝑋) < ((⌊‘(𝐵 logb 𝑋)) + 1)) | |
35 | 5, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 logb 𝑋) < ((⌊‘(𝐵 logb 𝑋)) + 1)) |
36 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐸 = (⌊‘(𝐵 logb 𝑋))) |
37 | 36 | oveq1d 6937 | . . . . 5 ⊢ (𝜑 → (𝐸 + 1) = ((⌊‘(𝐵 logb 𝑋)) + 1)) |
38 | 35, 37 | breqtrrd 4914 | . . . 4 ⊢ (𝜑 → (𝐵 logb 𝑋) < (𝐸 + 1)) |
39 | 16 | peano2zd 11837 | . . . . . 6 ⊢ (𝜑 → (𝐸 + 1) ∈ ℤ) |
40 | 39 | zred 11834 | . . . . 5 ⊢ (𝜑 → (𝐸 + 1) ∈ ℝ) |
41 | 11, 14, 5, 40 | cxpltd 24902 | . . . 4 ⊢ (𝜑 → ((𝐵 logb 𝑋) < (𝐸 + 1) ↔ (𝐵↑𝑐(𝐵 logb 𝑋)) < (𝐵↑𝑐(𝐸 + 1)))) |
42 | 38, 41 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐(𝐵 logb 𝑋)) < (𝐵↑𝑐(𝐸 + 1))) |
43 | 20, 23, 39 | cxpexpzd 24894 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐(𝐸 + 1)) = (𝐵↑(𝐸 + 1))) |
44 | 42, 32, 43 | 3brtr3d 4917 | . 2 ⊢ (𝜑 → 𝑋 < (𝐵↑(𝐸 + 1))) |
45 | 33, 44 | jca 507 | 1 ⊢ (𝜑 → ((𝐵↑𝐸) ≤ 𝑋 ∧ 𝑋 < (𝐵↑(𝐸 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∖ cdif 3788 {csn 4397 {cpr 4399 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 < clt 10411 ≤ cle 10412 ℕcn 11374 2c2 11430 ℤcz 11728 ℤ≥cuz 11992 ℝ+crp 12137 ⌊cfl 12910 ↑cexp 13178 ↑𝑐ccxp 24739 logb clogb 24942 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-sin 15202 df-cos 15203 df-pi 15205 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 df-cxp 24741 df-logb 24943 |
This theorem is referenced by: fldivexpfllog2 43367 |
Copyright terms: Public domain | W3C validator |