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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 12226 | . . . 4 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ) |
3 | blennnelnn 46652 | . . . 4 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ) | |
4 | nnm1nn0 12454 | . . . 4 ⊢ ((#b‘𝑁) ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℕ0) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℕ0) |
6 | nnre 12160 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
7 | nnnn0 12420 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
8 | 7 | nn0ge0d 12476 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
9 | elrege0 13371 | . . . 4 ⊢ (𝑁 ∈ (0[,)+∞) ↔ (𝑁 ∈ ℝ ∧ 0 ≤ 𝑁)) | |
10 | 6, 8, 9 | sylanbrc 583 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (0[,)+∞)) |
11 | nn0digval 46676 | . . 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 1371 | . 2 ⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1)(digit‘2)𝑁) = ((⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) mod 2)) |
13 | n2dvds1 16250 | . . . 4 ⊢ ¬ 2 ∥ 1 | |
14 | blennn 46651 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | |
15 | 14 | oveq1d 7372 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) = (((⌊‘(2 logb 𝑁)) + 1) − 1)) |
16 | 2z 12535 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ | |
17 | uzid 12778 | . . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ≥‘2) |
19 | nnrp 12926 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
20 | relogbzcl 26124 | . . . . . . . . . . . . . 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 13703 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℤ) |
23 | 22 | zcnd 12608 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℂ) |
24 | pncan1 11579 | . . . . . . . . . . 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 2776 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) = (⌊‘(2 logb 𝑁))) |
27 | 26 | oveq2d 7373 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) = (2↑(⌊‘(2 logb 𝑁)))) |
28 | 27 | oveq2d 7373 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 / (2↑((#b‘𝑁) − 1))) = (𝑁 / (2↑(⌊‘(2 logb 𝑁))))) |
29 | 28 | fveq2d 6846 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) = (⌊‘(𝑁 / (2↑(⌊‘(2 logb 𝑁)))))) |
30 | fldivexpfllog2 46641 | . . . . . . 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 2776 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) = 1) |
33 | 32 | breq2d 5117 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2 ∥ (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) ↔ 2 ∥ 1)) |
34 | 13, 33 | mtbiri 326 | . . 3 ⊢ (𝑁 ∈ ℕ → ¬ 2 ∥ (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1))))) |
35 | 2re 12227 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
36 | 35 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
37 | 36, 5 | reexpcld 14068 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ∈ ℝ) |
38 | 2cnd 12231 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
39 | 2ne0 12257 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
41 | 5 | nn0zd 12525 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℤ) |
42 | 38, 40, 41 | expne0d 14057 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ≠ 0) |
43 | 6, 37, 42 | redivcld 11983 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 / (2↑((#b‘𝑁) − 1))) ∈ ℝ) |
44 | 43 | flcld 13703 | . . . 4 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / (2↑((#b‘𝑁) − 1)))) ∈ ℤ) |
45 | mod2eq1n2dvds 16229 | . . . 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 2776 | 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 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 +∞cpnf 11186 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ0cn0 12413 ℤcz 12499 ℤ≥cuz 12763 ℝ+crp 12915 [,)cico 13266 ⌊cfl 13695 mod cmo 13774 ↑cexp 13967 ∥ cdvds 16136 logb clogb 26114 #bcblen 46645 digitcdig 46671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-dvds 16137 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-cxp 25913 df-logb 26115 df-blen 46646 df-dig 46672 |
This theorem is referenced by: (None) |
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