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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 12462 |
. . . 4
bits
    | |
| 2 | simp32 1058 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9765 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2322 |
. . . . . 6
..^
|
| 5 | simp1r 1046 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9575 |
. . . . . 6
..^
|
| 7 | 2re 9188 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 10920 |
. . . . . . . 8
..^
|
| 10 | simp1l 1045 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9577 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 10920 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8183 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8172 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9480 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 9829 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9280 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 10925 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 9827 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 411 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10474 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1059 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10471 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 411 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9232 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4091 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 9862 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9575 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 10927 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10360 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1043 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 9941 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9465 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10518 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 413 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12430 |
. . . . . . . . . . . . . . . . 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 8169 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 9950 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4107 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10361 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1043 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8279 |
. . . . . . 7
..^
|
| 58 | 1lt2 9288 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 10939 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1274 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10354 |
. . . . . 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 9131 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 534 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9431 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11731 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 411 |
. . . . . 6
bits ..^ |
| 74 | simplr 528 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12432 |
. . . . . . . . . . 11
| |
| 76 | simplll 533 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9577 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9430 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11744 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 10925 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9168 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8169 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9578 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9132 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9164 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 8958 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9431 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9131 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11730 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9482 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9744 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 10892 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 414 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8273 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8278 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8171 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4111 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 9898 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 9954 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4105 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8682 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9162 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9162 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9090 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4111 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8670 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 8100 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9223 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9229 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8299 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4125 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 9827 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 411 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9486 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10518 |
. . . . . . . . . . . . . 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 12463 |
. . . . . . . . . . 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 10361 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9575 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9575 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9500 |
. . . . . . . 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 9468 |
. . . . 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 12473 |
. 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 5318 (class class class)co 6007
csup 7157
infcinf 7158 cr 8006 cc0 8007 c1 8008 caddc 8010 cmul 8012 clt 8189 cle 8190 cneg 8326 cdiv 8827 cn 9118 c2 9169 cn0 9377 cz 9454 cuz 9730 cq 9822 crp 9857 ..^cfzo 10346
cfl 10496
cexp 10768
cdvds 12306 bitscbits 12459 |
| 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 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127 |
| 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 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-sup 7159 df-inf 7160 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-fz 10213 df-fzo 10347 df-fl 10498 df-seqfrec 10678 df-exp 10769 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-dvds 12307 df-bits 12460 |
| This theorem is referenced by: bitsfi 12476 0bits 12478 bitsinv1 12481 |
| Copyright terms: Public domain | W3C validator |