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 12745 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ+ | |
2 | 1 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ+) |
3 | nnrp 12751 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
4 | 1ne2 12191 | . . . . . . . . . . 11 ⊢ 1 ≠ 2 | |
5 | 4 | necomi 2998 | . . . . . . . . . 10 ⊢ 2 ≠ 1 |
6 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 2 ≠ 1) |
7 | relogbcl 25933 | . . . . . . . . 9 ⊢ ((2 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+ ∧ 2 ≠ 1) → (2 logb 𝑁) ∈ ℝ) | |
8 | 2, 3, 6, 7 | syl3anc 1370 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (2 logb 𝑁) ∈ ℝ) |
9 | 8 | flcld 13528 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℤ) |
10 | 9 | zcnd 12437 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℂ) |
11 | pncan1 11409 | . . . . . 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 7283 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑(((⌊‘(2 logb 𝑁)) + 1) − 1)) = (2↑(⌊‘(2 logb 𝑁)))) |
14 | blennn 45899 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | |
15 | 14 | oveq1d 7282 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) = (((⌊‘(2 logb 𝑁)) + 1) − 1)) |
16 | 15 | oveq2d 7283 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) = (2↑(((⌊‘(2 logb 𝑁)) + 1) − 1))) |
17 | 2cnd 12061 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
18 | 2ne0 12087 | . . . . . 6 ⊢ 2 ≠ 0 | |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
20 | 17, 19, 9 | cxpexpzd 25876 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑𝑐(⌊‘(2 logb 𝑁))) = (2↑(⌊‘(2 logb 𝑁)))) |
21 | 13, 16, 20 | 3eqtr4d 2788 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) = (2↑𝑐(⌊‘(2 logb 𝑁)))) |
22 | flle 13529 | . . . . . 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 12057 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
26 | 1lt2 12154 | . . . . . . 7 ⊢ 1 < 2 | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 < 2) |
28 | 9 | zred 12436 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℝ) |
29 | 25, 27, 28, 8 | cxpled 25885 | . . . . 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 12058 | . . . . . 6 ⊢ 2 ∈ ℂ | |
32 | eldifpr 4593 | . . . . . 6 ⊢ (2 ∈ (ℂ ∖ {0, 1}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1)) | |
33 | 31, 18, 5, 32 | mpbir3an 1340 | . . . . 5 ⊢ 2 ∈ (ℂ ∖ {0, 1}) |
34 | nncn 11991 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
35 | nnne0 12017 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
36 | eldifsn 4720 | . . . . . 6 ⊢ (𝑁 ∈ (ℂ ∖ {0}) ↔ (𝑁 ∈ ℂ ∧ 𝑁 ≠ 0)) | |
37 | 34, 35, 36 | sylanbrc 583 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℂ ∖ {0})) |
38 | cxplogb 25946 | . . . . 5 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑁 ∈ (ℂ ∖ {0})) → (2↑𝑐(2 logb 𝑁)) = 𝑁) | |
39 | 33, 37, 38 | sylancr 587 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑𝑐(2 logb 𝑁)) = 𝑁) |
40 | 30, 39 | breqtrd 5099 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑𝑐(⌊‘(2 logb 𝑁))) ≤ 𝑁) |
41 | 21, 40 | eqbrtrd 5095 | . 2 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ≤ 𝑁) |
42 | flltp1 13530 | . . . . . 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 12439 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((⌊‘(2 logb 𝑁)) + 1) ∈ ℤ) |
45 | 44 | zred 12436 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((⌊‘(2 logb 𝑁)) + 1) ∈ ℝ) |
46 | 25, 27, 8, 45 | cxpltd 25884 | . . . . 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 25876 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2↑𝑐((⌊‘(2 logb 𝑁)) + 1)) = (2↑((⌊‘(2 logb 𝑁)) + 1))) |
49 | 47, 39, 48 | 3brtr3d 5104 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2↑((⌊‘(2 logb 𝑁)) + 1))) |
50 | 14 | oveq2d 7283 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑(#b‘𝑁)) = (2↑((⌊‘(2 logb 𝑁)) + 1))) |
51 | 49, 50 | breqtrrd 5101 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2↑(#b‘𝑁))) |
52 | 41, 51 | jca 512 | 1 ⊢ (𝑁 ∈ ℕ → ((2↑((#b‘𝑁) − 1)) ≤ 𝑁 ∧ 𝑁 < (2↑(#b‘𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3883 {csn 4561 {cpr 4563 class class class wbr 5073 ‘cfv 6426 (class class class)co 7267 ℂcc 10879 ℝcr 10880 0cc0 10881 1c1 10882 + caddc 10884 < clt 11019 ≤ cle 11020 − cmin 11215 ℕcn 11983 2c2 12038 ℝ+crp 12740 ⌊cfl 13520 ↑cexp 13792 ↑𝑐ccxp 25721 logb clogb 25924 #bcblen 45893 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-fi 9157 df-sup 9188 df-inf 9189 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ioo 13093 df-ioc 13094 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-fl 13522 df-mod 13600 df-seq 13732 df-exp 13793 df-fac 13998 df-bc 14027 df-hash 14055 df-shft 14788 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-limsup 15190 df-clim 15207 df-rlim 15208 df-sum 15408 df-ef 15787 df-sin 15789 df-cos 15790 df-pi 15792 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-hom 16996 df-cco 16997 df-rest 17143 df-topn 17144 df-0g 17162 df-gsum 17163 df-topgen 17164 df-pt 17165 df-prds 17168 df-xrs 17223 df-qtop 17228 df-imas 17229 df-xps 17231 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-mulg 18711 df-cntz 18933 df-cmn 19398 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-fbas 20604 df-fg 20605 df-cnfld 20608 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cld 22180 df-ntr 22181 df-cls 22182 df-nei 22259 df-lp 22297 df-perf 22298 df-cn 22388 df-cnp 22389 df-haus 22476 df-tx 22723 df-hmeo 22916 df-fil 23007 df-fm 23099 df-flim 23100 df-flf 23101 df-xms 23483 df-ms 23484 df-tms 23485 df-cncf 24051 df-limc 25040 df-dv 25041 df-log 25722 df-cxp 25723 df-logb 25925 df-blen 45894 |
This theorem is referenced by: nnpw2blenfzo 45905 nnpw2pmod 45907 |
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