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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnpw2blen | Structured version Visualization version GIF version |
Description: A positive integer is between 2 to the power of its binary length minus 1 and 2 to the power of its binary length. (Contributed by AV, 31-May-2020.) |
Ref | Expression |
---|---|
nnpw2blen | ⊢ (𝑁 ∈ ℕ → ((2↑((#b‘𝑁) − 1)) ≤ 𝑁 ∧ 𝑁 < (2↑(#b‘𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2rp 12927 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ+ | |
2 | 1 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ+) |
3 | nnrp 12933 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
4 | 1ne2 12368 | . . . . . . . . . . 11 ⊢ 1 ≠ 2 | |
5 | 4 | necomi 2999 | . . . . . . . . . 10 ⊢ 2 ≠ 1 |
6 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 2 ≠ 1) |
7 | relogbcl 26139 | . . . . . . . . 9 ⊢ ((2 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+ ∧ 2 ≠ 1) → (2 logb 𝑁) ∈ ℝ) | |
8 | 2, 3, 6, 7 | syl3anc 1372 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (2 logb 𝑁) ∈ ℝ) |
9 | 8 | flcld 13710 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℤ) |
10 | 9 | zcnd 12615 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℂ) |
11 | pncan1 11586 | . . . . . 6 ⊢ ((⌊‘(2 logb 𝑁)) ∈ ℂ → (((⌊‘(2 logb 𝑁)) + 1) − 1) = (⌊‘(2 logb 𝑁))) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (((⌊‘(2 logb 𝑁)) + 1) − 1) = (⌊‘(2 logb 𝑁))) |
13 | 12 | oveq2d 7378 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑(((⌊‘(2 logb 𝑁)) + 1) − 1)) = (2↑(⌊‘(2 logb 𝑁)))) |
14 | blennn 46735 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | |
15 | 14 | oveq1d 7377 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) = (((⌊‘(2 logb 𝑁)) + 1) − 1)) |
16 | 15 | oveq2d 7378 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) = (2↑(((⌊‘(2 logb 𝑁)) + 1) − 1))) |
17 | 2cnd 12238 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
18 | 2ne0 12264 | . . . . . 6 ⊢ 2 ≠ 0 | |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
20 | 17, 19, 9 | cxpexpzd 26082 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑𝑐(⌊‘(2 logb 𝑁))) = (2↑(⌊‘(2 logb 𝑁)))) |
21 | 13, 16, 20 | 3eqtr4d 2787 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) = (2↑𝑐(⌊‘(2 logb 𝑁)))) |
22 | flle 13711 | . . . . . 6 ⊢ ((2 logb 𝑁) ∈ ℝ → (⌊‘(2 logb 𝑁)) ≤ (2 logb 𝑁)) | |
23 | 8, 22 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ≤ (2 logb 𝑁)) |
24 | 2re 12234 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
26 | 1lt2 12331 | . . . . . . 7 ⊢ 1 < 2 | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 < 2) |
28 | 9 | zred 12614 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℝ) |
29 | 25, 27, 28, 8 | cxpled 26091 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((⌊‘(2 logb 𝑁)) ≤ (2 logb 𝑁) ↔ (2↑𝑐(⌊‘(2 logb 𝑁))) ≤ (2↑𝑐(2 logb 𝑁)))) |
30 | 23, 29 | mpbid 231 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑𝑐(⌊‘(2 logb 𝑁))) ≤ (2↑𝑐(2 logb 𝑁))) |
31 | 2cn 12235 | . . . . . 6 ⊢ 2 ∈ ℂ | |
32 | eldifpr 4623 | . . . . . 6 ⊢ (2 ∈ (ℂ ∖ {0, 1}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1)) | |
33 | 31, 18, 5, 32 | mpbir3an 1342 | . . . . 5 ⊢ 2 ∈ (ℂ ∖ {0, 1}) |
34 | nncn 12168 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
35 | nnne0 12194 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
36 | eldifsn 4752 | . . . . . 6 ⊢ (𝑁 ∈ (ℂ ∖ {0}) ↔ (𝑁 ∈ ℂ ∧ 𝑁 ≠ 0)) | |
37 | 34, 35, 36 | sylanbrc 584 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℂ ∖ {0})) |
38 | cxplogb 26152 | . . . . 5 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑁 ∈ (ℂ ∖ {0})) → (2↑𝑐(2 logb 𝑁)) = 𝑁) | |
39 | 33, 37, 38 | sylancr 588 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑𝑐(2 logb 𝑁)) = 𝑁) |
40 | 30, 39 | breqtrd 5136 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑𝑐(⌊‘(2 logb 𝑁))) ≤ 𝑁) |
41 | 21, 40 | eqbrtrd 5132 | . 2 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ≤ 𝑁) |
42 | flltp1 13712 | . . . . . 6 ⊢ ((2 logb 𝑁) ∈ ℝ → (2 logb 𝑁) < ((⌊‘(2 logb 𝑁)) + 1)) | |
43 | 8, 42 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2 logb 𝑁) < ((⌊‘(2 logb 𝑁)) + 1)) |
44 | 9 | peano2zd 12617 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((⌊‘(2 logb 𝑁)) + 1) ∈ ℤ) |
45 | 44 | zred 12614 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((⌊‘(2 logb 𝑁)) + 1) ∈ ℝ) |
46 | 25, 27, 8, 45 | cxpltd 26090 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((2 logb 𝑁) < ((⌊‘(2 logb 𝑁)) + 1) ↔ (2↑𝑐(2 logb 𝑁)) < (2↑𝑐((⌊‘(2 logb 𝑁)) + 1)))) |
47 | 43, 46 | mpbid 231 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑𝑐(2 logb 𝑁)) < (2↑𝑐((⌊‘(2 logb 𝑁)) + 1))) |
48 | 17, 19, 44 | cxpexpzd 26082 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑𝑐((⌊‘(2 logb 𝑁)) + 1)) = (2↑((⌊‘(2 logb 𝑁)) + 1))) |
49 | 47, 39, 48 | 3brtr3d 5141 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2↑((⌊‘(2 logb 𝑁)) + 1))) |
50 | 14 | oveq2d 7378 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑(#b‘𝑁)) = (2↑((⌊‘(2 logb 𝑁)) + 1))) |
51 | 49, 50 | breqtrrd 5138 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2↑(#b‘𝑁))) |
52 | 41, 51 | jca 513 | 1 ⊢ (𝑁 ∈ ℕ → ((2↑((#b‘𝑁) − 1)) ≤ 𝑁 ∧ 𝑁 < (2↑(#b‘𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∖ cdif 3912 {csn 4591 {cpr 4593 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ℂcc 11056 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 ≤ cle 11197 − cmin 11392 ℕcn 12160 2c2 12215 ℝ+crp 12922 ⌊cfl 13702 ↑cexp 13974 ↑𝑐ccxp 25927 logb clogb 26130 #bcblen 46729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14959 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-ef 15957 df-sin 15959 df-cos 15960 df-pi 15962 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-mulg 18880 df-cntz 19104 df-cmn 19571 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 df-cxp 25929 df-logb 26131 df-blen 46730 |
This theorem is referenced by: nnpw2blenfzo 46741 nnpw2pmod 46743 |
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