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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 12125 |
. . . 4
bits
    | |
| 2 | simp32 1036 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9653 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2289 |
. . . . . 6
..^
|
| 5 | simp1r 1024 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9463 |
. . . . . 6
..^
|
| 7 | 2re 9077 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 10799 |
. . . . . . . 8
..^
|
| 10 | simp1l 1023 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9465 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 10799 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8072 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8061 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9369 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 9717 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9169 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 10804 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 9715 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 411 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10353 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1037 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10350 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 411 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9121 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4042 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 9750 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9463 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 10806 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10248 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1021 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 9829 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9354 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10397 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 413 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12093 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4072 |
. . . . . . . . . . . . . . . 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 664 |
. . . . . . . . . . 11
..^
|
| 49 | notnotrdc 844 |
. . . . . . . . . . 11
DECID
| |
| 50 | 25, 48, 49 | sylc 62 |
. . . . . . . . . 10
..^
|
| 51 | 1red 8058 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 9838 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4058 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10249 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1021 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8168 |
. . . . . . 7
..^
|
| 58 | 1lt2 9177 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 10818 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1252 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10242 |
. . . . . 6
..^
| |
| 64 | 4, 6, 62, 63 | syl3anbrc 1183 |
. . . . 5
..^
..^ |
| 65 | 64 | 3expia 1207 |
. . . 4
..^
..^ |
| 66 | 1, 65 | biimtrid 152 |
. . 3
..^ bits ..^ |
| 67 | 66 | ssrdv 3190 |
. 2
..^ bits
..^ |
| 68 | simpr 110 |
. . . . . . . 8
bits ..^  | |
| 69 | 68 | nnred 9020 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 534 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9320 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11394 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 411 |
. . . . . 6
bits ..^ |
| 74 | simplr 528 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12095 |
. . . . . . . . . . 11
| |
| 76 | simplll 533 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9465 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9319 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11407 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 10804 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9057 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8058 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9466 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9021 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9053 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 8847 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9320 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9020 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11393 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9371 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9632 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 10771 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 414 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8162 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8167 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8060 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4062 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 9786 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 9842 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4056 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8571 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9051 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9051 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 8979 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4062 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8559 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 7989 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9112 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9118 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8188 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4076 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 9715 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 411 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9375 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10397 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 413 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 946 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4046 |
. . . . . . . . . . 11
bits ..^
|
| 125 | 75, 124 | mtbiri 676 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12126 |
. . . . . . . . . . 11
bits
| |
| 127 | 76, 81, 126 | syl2anc 411 |
. . . . . . . . . 10
bits ..^
bits
|
| 128 | 125, 127 | mpbird 167 |
. . . . . . . . 9
bits ..^
bits |
| 129 | 74, 128 | sseldd 3185 |
. . . . . . . 8
bits ..^
..^ |
| 130 | elfzolt2 10249 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9463 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9463 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9389 |
. . . . . . . 8
| |
| 135 | 132, 133, 134 | syl2anc 411 |
. . . . . . 7
bits ..^
|
| 136 | 131, 135 | mpbid 147 |
. . . . . 6
bits ..^
|
| 137 | 73, 136 | pm2.65da 662 |
. . . . 5
bits ..^
|
| 138 | 137 | intnand 932 |
. . . 4
bits ..^
|
| 139 | simpll 527 |
. . . . 5
bits ..^
| |
| 140 | elznn0nn 9357 |
. . . . 5
 | |
| 141 | 139, 140 | sylib 122 |
. . . 4
bits ..^
|
| 142 | 138, 141 | ecased 1360 |
. . 3
bits ..^
|
| 143 | simplr 528 |
. . 3
bits ..^
| |
| 144 | simpr 110 |
. . 3
bits ..^
bits
..^ | |
| 145 | eqid 2196 |
. . 3
inf 
inf
| |
| 146 | 142, 143, 144, 145 | bitsfzolem 12136 |
. 2
bits ..^
..^ |
| 147 | 67, 146 | impbida 596 |
1
..^
bits
..^ |
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 104
wb 105 wo 709 DECID wdc 835
w3a 980
wceq 1364
wcel 2167
crab 2479
wss 3157
cpr 3624
class class class wbr 4034 cfv 5259 (class class class)co 5925
csup 7057
infcinf 7058 cr 7895 cc0 7896 c1 7897 caddc 7899 cmul 7901 clt 8078 cle 8079 cneg 8215 cdiv 8716 cn 9007 c2 9058 cn0 9266 cz 9343 cuz 9618 cq 9710 crp 9745 ..^cfzo 10234
cfl 10375
cexp 10647
cdvds 11969 bitscbits 12122 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-fz 10101 df-fzo 10235 df-fl 10377 df-seqfrec 10557 df-exp 10648 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-dvds 11970 df-bits 12123 |
| This theorem is referenced by: bitsfi 12139 0bits 12141 bitsinv1 12144 |
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