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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 12127 |
. . . 4
bits
    | |
| 2 | simp32 1036 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9655 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2289 |
. . . . . 6
..^
|
| 5 | simp1r 1024 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9465 |
. . . . . 6
..^
|
| 7 | 2re 9079 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 10801 |
. . . . . . . 8
..^
|
| 10 | simp1l 1023 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9467 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 10801 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8074 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8063 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9371 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 9719 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9171 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 10806 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 9717 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 411 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10355 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1037 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10352 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 411 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9123 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4042 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 9752 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9465 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 10808 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10250 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1021 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 9831 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9356 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10399 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 413 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12095 |
. . . . . . . . . . . . . . . . 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 8060 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 9840 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4058 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10251 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1021 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8170 |
. . . . . . 7
..^
|
| 58 | 1lt2 9179 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 10820 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1252 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10244 |
. . . . . 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 9022 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 534 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9322 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11396 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 411 |
. . . . . 6
bits ..^ |
| 74 | simplr 528 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12097 |
. . . . . . . . . . 11
| |
| 76 | simplll 533 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9467 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9321 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11409 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 10806 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9059 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8060 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9468 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9023 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9055 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 8849 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9322 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9022 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11395 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9373 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9634 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 10773 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 414 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8164 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8169 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8062 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4062 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 9788 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 9844 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4056 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8573 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9053 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9053 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 8981 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4062 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8561 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 7991 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9114 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9120 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8190 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4076 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 9717 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 411 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9377 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10399 |
. . . . . . . . . . . . . 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 12128 |
. . . . . . . . . . 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 10251 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9465 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9465 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9391 |
. . . . . . . 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 9359 |
. . . . 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 12138 |
. 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 7897 cc0 7898 c1 7899 caddc 7901 cmul 7903 clt 8080 cle 8081 cneg 8217 cdiv 8718 cn 9009 c2 9060 cn0 9268 cz 9345 cuz 9620 cq 9712 crp 9747 ..^cfzo 10236
cfl 10377
cexp 10649
cdvds 11971 bitscbits 12124 |
| 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 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017 ax-caucvg 8018 |
| 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 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-n0 9269 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-fz 10103 df-fzo 10237 df-fl 10379 df-seqfrec 10559 df-exp 10650 df-cj 11026 df-re 11027 df-im 11028 df-rsqrt 11182 df-abs 11183 df-dvds 11972 df-bits 12125 |
| This theorem is referenced by: bitsfi 12141 0bits 12143 bitsinv1 12146 |
| Copyright terms: Public domain | W3C validator |