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| Mirrors > Home > ILE Home > Th. List > bitsfzo | Unicode version | ||
Description: The bits of a number are
all at positions less than  iff the number
is nonnegative and less than   . (Contributed by Mario
Carneiro, 5-Sep-2016.) (Proof shortened by AV,
1-Oct-2020.) |
| Ref | Expression |
|---|---|
| bitsfzo |
     ![/\](wedge.gif "/\"){kind=small}
    ..^ 
bits 
..^ |
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bitsval 12627 |
. . . 4
bits
    | |
| 2 | simp32 1061 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9889 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2325 |
. . . . . 6
..^
|
| 5 | simp1r 1049 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9698 |
. . . . . 6
..^
|
| 7 | 2re 9307 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 11052 |
. . . . . . . 8
..^
|
| 10 | simp1l 1048 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9700 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 11052 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8302 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8292 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9603 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 9958 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9399 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 11057 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 9956 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 411 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10606 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1062 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10603 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 411 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9351 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4117 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 9991 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9698 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 11059 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10490 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1046 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 10070 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9588 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10650 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 413 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12595 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4147 |
. . . . . . . . . . . . . . . 16
|
| 44 | 40, 43 | biimtrrdi 164 |
. . . . . . . . . . . . . . 15
..^
|
| 45 | 37, 44 | mpand 429 |
. . . . . . . . . . . . . 14
..^
|
| 46 | 30, 45 | biimtrrid 153 |
. . . . . . . . . . . . 13
..^
|
| 47 | 28, 46 | sylbird 170 |
. . . . . . . . . . . 12
..^
|
| 48 | 26, 47 | mtod 669 |
. . . . . . . . . . 11
..^
|
| 49 | notnotrdc 851 |
. . . . . . . . . . 11
DECID
| |
| 50 | 25, 48, 49 | sylc 62 |
. . . . . . . . . 10
..^
|
| 51 | 1red 8289 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 10079 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4133 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10491 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1046 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8398 |
. . . . . . 7
..^
|
| 58 | 1lt2 9407 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 11071 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1277 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10484 |
. . . . . 6
..^
| |
| 64 | 4, 6, 62, 63 | syl3anbrc 1208 |
. . . . 5
..^
..^ |
| 65 | 64 | 3expia 1232 |
. . . 4
..^
..^ |
| 66 | 1, 65 | biimtrid 152 |
. . 3
..^ bits ..^ |
| 67 | 66 | ssrdv 3244 |
. 2
..^ bits
..^ |
| 68 | simpr 110 |
. . . . . . . 8
bits ..^  | |
| 69 | 68 | nnred 9250 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 536 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9554 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11895 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 411 |
. . . . . 6
bits ..^ |
| 74 | simplr 529 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12597 |
. . . . . . . . . . 11
| |
| 76 | simplll 535 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9700 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9553 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11908 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 11057 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9287 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8289 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9701 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9251 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9283 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 9077 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9554 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9250 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11894 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9605 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9868 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 11024 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 414 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8392 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8397 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8291 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4137 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 10027 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 10083 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4131 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8801 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9281 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9281 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9209 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4137 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8789 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 8220 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9342 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9348 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8418 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4151 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 9956 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 411 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9609 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10650 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 413 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 953 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4121 |
. . . . . . . . . . 11
bits ..^
|
| 125 | 75, 124 | mtbiri 682 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12628 |
. . . . . . . . . . 11
bits
| |
| 127 | 76, 81, 126 | syl2anc 411 |
. . . . . . . . . 10
bits ..^
bits
|
| 128 | 125, 127 | mpbird 167 |
. . . . . . . . 9
bits ..^
bits |
| 129 | 74, 128 | sseldd 3239 |
. . . . . . . 8
bits ..^
..^ |
| 130 | elfzolt2 10491 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9698 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9698 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9623 |
. . . . . . . 8
| |
| 135 | 132, 133, 134 | syl2anc 411 |
. . . . . . 7
bits ..^
|
| 136 | 131, 135 | mpbid 147 |
. . . . . 6
bits ..^
|
| 137 | 73, 136 | pm2.65da 667 |
. . . . 5
bits ..^
|
| 138 | 137 | intnand 939 |
. . . 4
bits ..^
|
| 139 | simpll 527 |
. . . . 5
bits ..^
| |
| 140 | elznn0nn 9591 |
. . . . 5
 | |
| 141 | 139, 140 | sylib 122 |
. . . 4
bits ..^
|
| 142 | 138, 141 | ecased 1386 |
. . 3
bits ..^
|
| 143 | simplr 529 |
. . 3
bits ..^
| |
| 144 | simpr 110 |
. . 3
bits ..^
bits
..^ | |
| 145 | eqid 2232 |
. . 3
inf 
inf
| |
| 146 | 142, 143, 144, 145 | bitsfzolem 12638 |
. 2
bits ..^
..^ |
| 147 | 67, 146 | impbida 600 |
1
..^
bits
..^ |
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 104
wb 105 wo 716 DECID wdc 842
w3a 1005
wceq 1398
wcel 2203
crab 2524
wss 3211
cpr 3690
class class class wbr 4109 cfv 5352 (class class class)co 6050
csup 7273
infcinf 7274 cr 8126 cc0 8127 c1 8128 caddc 8130 cmul 8132 clt 8308 cle 8309 cneg 8445 cdiv 8946 cn 9237 c2 9288 cn0 9496 cz 9577 cuz 9853 cq 9951 crp 9986 ..^cfzo 10476
cfl 10628
cexp 10900
cdvds 12471 bitscbits 12624 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247 |
| This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-sup 7275 df-inf 7276 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-fz 10343 df-fzo 10477 df-fl 10630 df-seqfrec 10810 df-exp 10901 df-cj 11525 df-re 11526 df-im 11527 df-rsqrt 11681 df-abs 11682 df-dvds 12472 df-bits 12625 |
| This theorem is referenced by: bitsfi 12641 0bits 12643 bitsinv1 12646 |
| Copyright terms: Public domain | W3C validator |