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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 12046 | . . . 4 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ) |
3 | blennnelnn 45891 | . . . 4 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ) | |
4 | nnm1nn0 12274 | . . . 4 ⊢ ((#b‘𝑁) ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℕ0) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℕ0) |
6 | nnre 11980 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
7 | nnnn0 12240 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
8 | 7 | nn0ge0d 12296 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
9 | elrege0 13185 | . . . 4 ⊢ (𝑁 ∈ (0[,)+∞) ↔ (𝑁 ∈ ℝ ∧ 0 ≤ 𝑁)) | |
10 | 6, 8, 9 | sylanbrc 583 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (0[,)+∞)) |
11 | nn0digval 45915 | . . 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 1370 | . 2 ⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1)(digit‘2)𝑁) = ((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) mod 2)) |
13 | n2dvds1 16075 | . . . 4 ⊢ ¬ 2 ∥ 1 | |
14 | blennn 45890 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | |
15 | 14 | oveq1d 7286 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) = (((⌊‘(2 logb 𝑁)) + 1) − 1)) |
16 | 2z 12352 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ | |
17 | uzid 12596 | . . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ≥‘2) |
19 | nnrp 12740 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
20 | relogbzcl 25922 | . . . . . . . . . . . . . 14 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → (2 logb 𝑁) ∈ ℝ) | |
21 | 18, 19, 20 | sylancr 587 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (2 logb 𝑁) ∈ ℝ) |
22 | 21 | flcld 13516 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℤ) |
23 | 22 | zcnd 12426 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℂ) |
24 | pncan1 11399 | . . . . . . . . . . 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 2780 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) = (⌊‘(2 logb 𝑁))) |
27 | 26 | oveq2d 7287 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) = (2↑(⌊‘(2 logb 𝑁)))) |
28 | 27 | oveq2d 7287 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 / (2↑((#b‘𝑁) − 1))) = (𝑁 / (2↑(⌊‘(2 logb 𝑁))))) |
29 | 28 | fveq2d 6775 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) = (⌊‘(𝑁 / (2↑(⌊‘(2 logb 𝑁)))))) |
30 | fldivexpfllog2 45880 | . . . . . . 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 2780 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) = 1) |
33 | 32 | breq2d 5091 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2 ∥ (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) ↔ 2 ∥ 1)) |
34 | 13, 33 | mtbiri 327 | . . 3 ⊢ (𝑁 ∈ ℕ → ¬ 2 ∥ (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1))))) |
35 | 2re 12047 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
36 | 35 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
37 | 36, 5 | reexpcld 13879 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ∈ ℝ) |
38 | 2cnd 12051 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
39 | 2ne0 12077 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
41 | 5 | nn0zd 12423 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℤ) |
42 | 38, 40, 41 | expne0d 13868 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ≠ 0) |
43 | 6, 37, 42 | redivcld 11803 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 / (2↑((#b‘𝑁) − 1))) ∈ ℝ) |
44 | 43 | flcld 13516 | . . . 4 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) ∈ ℤ) |
45 | mod2eq1n2dvds 16054 | . . . 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 256 | . 2 ⊢ (𝑁 ∈ ℕ → ((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) mod 2) = 1) |
48 | 12, 47 | eqtrd 2780 | 1 ⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1)(digit‘2)𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 class class class wbr 5079 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 ℝcr 10871 0cc0 10872 1c1 10873 + caddc 10875 +∞cpnf 11007 ≤ cle 11011 − cmin 11205 / cdiv 11632 ℕcn 11973 2c2 12028 ℕ0cn0 12233 ℤcz 12319 ℤ≥cuz 12581 ℝ+crp 12729 [,)cico 13080 ⌊cfl 13508 mod cmo 13587 ↑cexp 13780 ∥ cdvds 15961 logb clogb 25912 #bcblen 45884 digitcdig 45910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ioc 13083 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-fac 13986 df-bc 14015 df-hash 14043 df-shft 14776 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-limsup 15178 df-clim 15195 df-rlim 15196 df-sum 15396 df-ef 15775 df-sin 15777 df-cos 15778 df-pi 15780 df-dvds 15962 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cncf 24039 df-limc 25028 df-dv 25029 df-log 25710 df-cxp 25711 df-logb 25913 df-blen 45885 df-dig 45911 |
This theorem is referenced by: (None) |
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