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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 12475 |
. . . 4
bits
    | |
| 2 | simp32 1058 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9774 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2322 |
. . . . . 6
..^
|
| 5 | simp1r 1046 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9583 |
. . . . . 6
..^
|
| 7 | 2re 9196 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 10929 |
. . . . . . . 8
..^
|
| 10 | simp1l 1045 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9585 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 10929 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8191 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8180 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9488 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 9838 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9288 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 10934 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 9836 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 411 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10483 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1059 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10480 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 411 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9240 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4091 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 9871 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9583 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 10936 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10369 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1043 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 9950 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9473 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10527 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 413 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12443 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4121 |
. . . . . . . . . . . . . . . 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 667 |
. . . . . . . . . . 11
..^
|
| 49 | notnotrdc 848 |
. . . . . . . . . . 11
DECID
| |
| 50 | 25, 48, 49 | sylc 62 |
. . . . . . . . . 10
..^
|
| 51 | 1red 8177 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 9959 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4107 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10370 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1043 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8287 |
. . . . . . 7
..^
|
| 58 | 1lt2 9296 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 10948 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1274 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10363 |
. . . . . 6
..^
| |
| 64 | 4, 6, 62, 63 | syl3anbrc 1205 |
. . . . 5
..^
..^ |
| 65 | 64 | 3expia 1229 |
. . . 4
..^
..^ |
| 66 | 1, 65 | biimtrid 152 |
. . 3
..^ bits ..^ |
| 67 | 66 | ssrdv 3230 |
. 2
..^ bits
..^ |
| 68 | simpr 110 |
. . . . . . . 8
bits ..^  | |
| 69 | 68 | nnred 9139 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 534 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9439 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11744 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 411 |
. . . . . 6
bits ..^ |
| 74 | simplr 528 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12445 |
. . . . . . . . . . 11
| |
| 76 | simplll 533 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9585 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9438 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11757 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 10934 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9176 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8177 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9586 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9140 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9172 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 8966 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9439 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9139 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11743 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9490 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9753 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 10901 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 414 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8281 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8286 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8179 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4111 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 9907 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 9963 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4105 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8690 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9170 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9170 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9098 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4111 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8678 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 8108 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9231 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9237 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8307 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4125 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 9836 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 411 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9494 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10527 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 413 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 950 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4095 |
. . . . . . . . . . 11
bits ..^
|
| 125 | 75, 124 | mtbiri 679 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12476 |
. . . . . . . . . . 11
bits
| |
| 127 | 76, 81, 126 | syl2anc 411 |
. . . . . . . . . 10
bits ..^
bits
|
| 128 | 125, 127 | mpbird 167 |
. . . . . . . . 9
bits ..^
bits |
| 129 | 74, 128 | sseldd 3225 |
. . . . . . . 8
bits ..^
..^ |
| 130 | elfzolt2 10370 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9583 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9583 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9508 |
. . . . . . . 8
| |
| 135 | 132, 133, 134 | syl2anc 411 |
. . . . . . 7
bits ..^
|
| 136 | 131, 135 | mpbid 147 |
. . . . . 6
bits ..^
|
| 137 | 73, 136 | pm2.65da 665 |
. . . . 5
bits ..^
|
| 138 | 137 | intnand 936 |
. . . 4
bits ..^
|
| 139 | simpll 527 |
. . . . 5
bits ..^
| |
| 140 | elznn0nn 9476 |
. . . . 5
 | |
| 141 | 139, 140 | sylib 122 |
. . . 4
bits ..^
|
| 142 | 138, 141 | ecased 1383 |
. . 3
bits ..^
|
| 143 | simplr 528 |
. . 3
bits ..^
| |
| 144 | simpr 110 |
. . 3
bits ..^
bits
..^ | |
| 145 | eqid 2229 |
. . 3
inf 
inf
| |
| 146 | 142, 143, 144, 145 | bitsfzolem 12486 |
. 2
bits ..^
..^ |
| 147 | 67, 146 | impbida 598 |
1
..^
bits
..^ |
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 104
wb 105 wo 713 DECID wdc 839
w3a 1002
wceq 1395
wcel 2200
crab 2512
wss 3197
cpr 3667
class class class wbr 4083 cfv 5321 (class class class)co 6010
csup 7165
infcinf 7166 cr 8014 cc0 8015 c1 8016 caddc 8018 cmul 8020 clt 8197 cle 8198 cneg 8334 cdiv 8835 cn 9126 c2 9177 cn0 9385 cz 9462 cuz 9738 cq 9831 crp 9866 ..^cfzo 10355
cfl 10505
cexp 10777
cdvds 12319 bitscbits 12472 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133 ax-arch 8134 ax-caucvg 8135 |
| This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-isom 5330 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-1st 6295 df-2nd 6296 df-recs 6462 df-frec 6548 df-sup 7167 df-inf 7168 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836 df-inn 9127 df-2 9185 df-3 9186 df-4 9187 df-n0 9386 df-z 9463 df-uz 9739 df-q 9832 df-rp 9867 df-fz 10222 df-fzo 10356 df-fl 10507 df-seqfrec 10687 df-exp 10778 df-cj 11374 df-re 11375 df-im 11376 df-rsqrt 11530 df-abs 11531 df-dvds 12320 df-bits 12473 |
| This theorem is referenced by: bitsfi 12489 0bits 12491 bitsinv1 12494 |
| Copyright terms: Public domain | W3C validator |