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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dig2nn1st | Structured version Visualization version GIF version |
Description: The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.) |
Ref | Expression |
---|---|
dig2nn1st | ⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1)(digit‘2)𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 11397 | . . . 4 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ) |
3 | blennnelnn 42898 | . . . 4 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ) | |
4 | nnm1nn0 11546 | . . . 4 ⊢ ((#b‘𝑁) ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℕ0) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℕ0) |
6 | nnre 11239 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
7 | nnnn0 11511 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
8 | 7 | nn0ge0d 11566 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
9 | elrege0 12491 | . . . 4 ⊢ (𝑁 ∈ (0[,)+∞) ↔ (𝑁 ∈ ℝ ∧ 0 ≤ 𝑁)) | |
10 | 6, 8, 9 | sylanbrc 701 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (0[,)+∞)) |
11 | nn0digval 42922 | . . 3 ⊢ ((2 ∈ ℕ ∧ ((#b‘𝑁) − 1) ∈ ℕ0 ∧ 𝑁 ∈ (0[,)+∞)) → (((#b‘𝑁) − 1)(digit‘2)𝑁) = ((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) mod 2)) | |
12 | 2, 5, 10, 11 | syl3anc 1477 | . 2 ⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1)(digit‘2)𝑁) = ((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) mod 2)) |
13 | n2dvds1 15326 | . . . 4 ⊢ ¬ 2 ∥ 1 | |
14 | blennn 42897 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | |
15 | 14 | oveq1d 6829 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) = (((⌊‘(2 logb 𝑁)) + 1) − 1)) |
16 | 2z 11621 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ | |
17 | uzid 11914 | . . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ≥‘2) |
19 | nnrp 12055 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
20 | relogbzcl 24732 | . . . . . . . . . . . . . 14 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → (2 logb 𝑁) ∈ ℝ) | |
21 | 18, 19, 20 | sylancr 698 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (2 logb 𝑁) ∈ ℝ) |
22 | 21 | flcld 12813 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℤ) |
23 | 22 | zcnd 11695 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℂ) |
24 | pncan1 10666 | . . . . . . . . . . 11 ⊢ ((⌊‘(2 logb 𝑁)) ∈ ℂ → (((⌊‘(2 logb 𝑁)) + 1) − 1) = (⌊‘(2 logb 𝑁))) | |
25 | 23, 24 | syl 17 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((⌊‘(2 logb 𝑁)) + 1) − 1) = (⌊‘(2 logb 𝑁))) |
26 | 15, 25 | eqtrd 2794 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) = (⌊‘(2 logb 𝑁))) |
27 | 26 | oveq2d 6830 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) = (2↑(⌊‘(2 logb 𝑁)))) |
28 | 27 | oveq2d 6830 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 / (2↑((#b‘𝑁) − 1))) = (𝑁 / (2↑(⌊‘(2 logb 𝑁))))) |
29 | 28 | fveq2d 6357 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) = (⌊‘(𝑁 / (2↑(⌊‘(2 logb 𝑁)))))) |
30 | fldivexpfllog2 42887 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ+ → (⌊‘(𝑁 / (2↑(⌊‘(2 logb 𝑁))))) = 1) | |
31 | 19, 30 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑(⌊‘(2 logb 𝑁))))) = 1) |
32 | 29, 31 | eqtrd 2794 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) = 1) |
33 | 32 | breq2d 4816 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2 ∥ (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) ↔ 2 ∥ 1)) |
34 | 13, 33 | mtbiri 316 | . . 3 ⊢ (𝑁 ∈ ℕ → ¬ 2 ∥ (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1))))) |
35 | 2re 11302 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
36 | 35 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
37 | 36, 5 | reexpcld 13239 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ∈ ℝ) |
38 | 2cnd 11305 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
39 | 2ne0 11325 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
41 | 5 | nn0zd 11692 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℤ) |
42 | 38, 40, 41 | expne0d 13228 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ≠ 0) |
43 | 6, 37, 42 | redivcld 11065 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 / (2↑((#b‘𝑁) − 1))) ∈ ℝ) |
44 | 43 | flcld 12813 | . . . 4 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) ∈ ℤ) |
45 | mod2eq1n2dvds 15293 | . . . 4 ⊢ ((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) ∈ ℤ → (((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) mod 2) = 1 ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))))) | |
46 | 44, 45 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → (((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) mod 2) = 1 ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))))) |
47 | 34, 46 | mpbird 247 | . 2 ⊢ (𝑁 ∈ ℕ → ((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) mod 2) = 1) |
48 | 12, 47 | eqtrd 2794 | 1 ⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1)(digit‘2)𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 ℝcr 10147 0cc0 10148 1c1 10149 + caddc 10151 +∞cpnf 10283 ≤ cle 10287 − cmin 10478 / cdiv 10896 ℕcn 11232 2c2 11282 ℕ0cn0 11504 ℤcz 11589 ℤ≥cuz 11899 ℝ+crp 12045 [,)cico 12390 ⌊cfl 12805 mod cmo 12882 ↑cexp 13074 ∥ cdvds 15202 logb clogb 24722 #bcblen 42891 digitcdig 42917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-fi 8484 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-ioo 12392 df-ioc 12393 df-ico 12394 df-icc 12395 df-fz 12540 df-fzo 12680 df-fl 12807 df-mod 12883 df-seq 13016 df-exp 13075 df-fac 13275 df-bc 13304 df-hash 13332 df-shft 14026 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-limsup 14421 df-clim 14438 df-rlim 14439 df-sum 14636 df-ef 15017 df-sin 15019 df-cos 15020 df-pi 15022 df-dvds 15203 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-hom 16188 df-cco 16189 df-rest 16305 df-topn 16306 df-0g 16324 df-gsum 16325 df-topgen 16326 df-pt 16327 df-prds 16330 df-xrs 16384 df-qtop 16389 df-imas 16390 df-xps 16392 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-submnd 17557 df-mulg 17762 df-cntz 17970 df-cmn 18415 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-fbas 19965 df-fg 19966 df-cnfld 19969 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-cld 21045 df-ntr 21046 df-cls 21047 df-nei 21124 df-lp 21162 df-perf 21163 df-cn 21253 df-cnp 21254 df-haus 21341 df-tx 21587 df-hmeo 21780 df-fil 21871 df-fm 21963 df-flim 21964 df-flf 21965 df-xms 22346 df-ms 22347 df-tms 22348 df-cncf 22902 df-limc 23849 df-dv 23850 df-log 24523 df-cxp 24524 df-logb 24723 df-blen 42892 df-dig 42918 |
This theorem is referenced by: (None) |
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