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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dig2bits | Structured version Visualization version GIF version |
Description: The πΎ th digit of a nonnegative integer π in a binary system is its πΎ th bit. (Contributed by AV, 24-May-2020.) |
Ref | Expression |
---|---|
dig2bits | β’ ((π β β0 β§ πΎ β β0) β ((πΎ(digitβ2)π) = 1 β πΎ β (bitsβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 12429 | . . . . . 6 β’ (π β β0 β π β β) | |
2 | 1 | adantr 482 | . . . . 5 β’ ((π β β0 β§ πΎ β β0) β π β β) |
3 | 2re 12234 | . . . . . . 7 β’ 2 β β | |
4 | 3 | a1i 11 | . . . . . 6 β’ (π β β0 β 2 β β) |
5 | reexpcl 13991 | . . . . . 6 β’ ((2 β β β§ πΎ β β0) β (2βπΎ) β β) | |
6 | 4, 5 | sylan 581 | . . . . 5 β’ ((π β β0 β§ πΎ β β0) β (2βπΎ) β β) |
7 | 2cnd 12238 | . . . . . 6 β’ ((π β β0 β§ πΎ β β0) β 2 β β) | |
8 | 2ne0 12264 | . . . . . . 7 β’ 2 β 0 | |
9 | 8 | a1i 11 | . . . . . 6 β’ ((π β β0 β§ πΎ β β0) β 2 β 0) |
10 | nn0z 12531 | . . . . . . 7 β’ (πΎ β β0 β πΎ β β€) | |
11 | 10 | adantl 483 | . . . . . 6 β’ ((π β β0 β§ πΎ β β0) β πΎ β β€) |
12 | 7, 9, 11 | expne0d 14064 | . . . . 5 β’ ((π β β0 β§ πΎ β β0) β (2βπΎ) β 0) |
13 | 2, 6, 12 | redivcld 11990 | . . . 4 β’ ((π β β0 β§ πΎ β β0) β (π / (2βπΎ)) β β) |
14 | 13 | flcld 13710 | . . 3 β’ ((π β β0 β§ πΎ β β0) β (ββ(π / (2βπΎ))) β β€) |
15 | mod2eq1n2dvds 16236 | . . 3 β’ ((ββ(π / (2βπΎ))) β β€ β (((ββ(π / (2βπΎ))) mod 2) = 1 β Β¬ 2 β₯ (ββ(π / (2βπΎ))))) | |
16 | 14, 15 | syl 17 | . 2 β’ ((π β β0 β§ πΎ β β0) β (((ββ(π / (2βπΎ))) mod 2) = 1 β Β¬ 2 β₯ (ββ(π / (2βπΎ))))) |
17 | 2nn 12233 | . . . . 5 β’ 2 β β | |
18 | 17 | a1i 11 | . . . 4 β’ ((π β β0 β§ πΎ β β0) β 2 β β) |
19 | simpr 486 | . . . 4 β’ ((π β β0 β§ πΎ β β0) β πΎ β β0) | |
20 | nn0rp0 13379 | . . . . 5 β’ (π β β0 β π β (0[,)+β)) | |
21 | 20 | adantr 482 | . . . 4 β’ ((π β β0 β§ πΎ β β0) β π β (0[,)+β)) |
22 | nn0digval 46760 | . . . 4 β’ ((2 β β β§ πΎ β β0 β§ π β (0[,)+β)) β (πΎ(digitβ2)π) = ((ββ(π / (2βπΎ))) mod 2)) | |
23 | 18, 19, 21, 22 | syl3anc 1372 | . . 3 β’ ((π β β0 β§ πΎ β β0) β (πΎ(digitβ2)π) = ((ββ(π / (2βπΎ))) mod 2)) |
24 | 23 | eqeq1d 2739 | . 2 β’ ((π β β0 β§ πΎ β β0) β ((πΎ(digitβ2)π) = 1 β ((ββ(π / (2βπΎ))) mod 2) = 1)) |
25 | nn0z 12531 | . . 3 β’ (π β β0 β π β β€) | |
26 | bitsval2 16312 | . . 3 β’ ((π β β€ β§ πΎ β β0) β (πΎ β (bitsβπ) β Β¬ 2 β₯ (ββ(π / (2βπΎ))))) | |
27 | 25, 26 | sylan 581 | . 2 β’ ((π β β0 β§ πΎ β β0) β (πΎ β (bitsβπ) β Β¬ 2 β₯ (ββ(π / (2βπΎ))))) |
28 | 16, 24, 27 | 3bitr4d 311 | 1 β’ ((π β β0 β§ πΎ β β0) β ((πΎ(digitβ2)π) = 1 β πΎ β (bitsβπ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 class class class wbr 5110 βcfv 6501 (class class class)co 7362 βcr 11057 0cc0 11058 1c1 11059 +βcpnf 11193 / cdiv 11819 βcn 12160 2c2 12215 β0cn0 12420 β€cz 12506 [,)cico 13273 βcfl 13702 mod cmo 13781 βcexp 13974 β₯ cdvds 16143 bitscbits 16306 digitcdig 46755 |
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-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 |
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-op 4598 df-uni 4871 df-iun 4961 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-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-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 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-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-ico 13277 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-dvds 16144 df-bits 16309 df-dig 46756 |
This theorem is referenced by: (None) |
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