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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 12728 |
. . . 4
bits
    | |
| 2 | simp32 1065 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9967 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2331 |
. . . . . 6
..^
|
| 5 | simp1r 1053 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9771 |
. . . . . 6
..^
|
| 7 | 2re 9377 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 11142 |
. . . . . . . 8
..^
|
| 10 | simp1l 1052 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9773 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 11142 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8355 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8345 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9675 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 10036 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9471 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 11147 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 10034 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 415 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10692 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 415 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1066 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10689 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 415 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9421 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4138 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 10070 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9771 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 11149 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10574 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1050 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 10149 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9660 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10739 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 417 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12696 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4168 |
. . . . . . . . . . . . . . . 16
|
| 44 | 40, 43 | biimtrrdi 164 |
. . . . . . . . . . . . . . 15
..^
|
| 45 | 37, 44 | mpand 433 |
. . . . . . . . . . . . . 14
..^
|
| 46 | 30, 45 | biimtrrid 153 |
. . . . . . . . . . . . 13
..^
|
| 47 | 28, 46 | sylbird 170 |
. . . . . . . . . . . 12
..^
|
| 48 | 26, 47 | mtod 673 |
. . . . . . . . . . 11
..^
|
| 49 | notnotrdc 855 |
. . . . . . . . . . 11
DECID
| |
| 50 | 25, 48, 49 | sylc 62 |
. . . . . . . . . 10
..^
|
| 51 | 1red 8342 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 10158 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4154 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10575 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1050 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8453 |
. . . . . . 7
..^
|
| 58 | 1lt2 9479 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 11162 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1281 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10568 |
. . . . . 6
..^
| |
| 64 | 4, 6, 62, 63 | syl3anbrc 1212 |
. . . . 5
..^
..^ |
| 65 | 64 | 3expia 1236 |
. . . 4
..^
..^ |
| 66 | 1, 65 | biimtrid 152 |
. . 3
..^ bits ..^ |
| 67 | 66 | ssrdv 3254 |
. 2
..^ bits
..^ |
| 68 | simpr 110 |
. . . . . . . 8
bits ..^  | |
| 69 | 68 | nnred 9320 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 540 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9626 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11994 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 415 |
. . . . . 6
bits ..^ |
| 74 | simplr 533 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12698 |
. . . . . . . . . . 11
| |
| 76 | simplll 539 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9773 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9625 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 12007 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 415 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 11147 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9357 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8342 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9774 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9321 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9353 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 9136 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9626 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9320 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11993 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 415 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9677 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9946 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 11114 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 418 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8447 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8452 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8344 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4158 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 10106 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 10162 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4152 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8857 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9351 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9351 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9268 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4158 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8845 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 8273 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9412 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9418 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8473 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4172 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 10034 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 415 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9681 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10739 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 417 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 957 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4142 |
. . . . . . . . . . 11
bits ..^
|
| 125 | 75, 124 | mtbiri 686 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12729 |
. . . . . . . . . . 11
bits
| |
| 127 | 76, 81, 126 | syl2anc 415 |
. . . . . . . . . 10
bits ..^
bits
|
| 128 | 125, 127 | mpbird 167 |
. . . . . . . . 9
bits ..^
bits |
| 129 | 74, 128 | sseldd 3249 |
. . . . . . . 8
bits ..^
..^ |
| 130 | elfzolt2 10575 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9771 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9771 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9695 |
. . . . . . . 8
| |
| 135 | 132, 133, 134 | syl2anc 415 |
. . . . . . 7
bits ..^
|
| 136 | 131, 135 | mpbid 147 |
. . . . . 6
bits ..^
|
| 137 | 73, 136 | pm2.65da 671 |
. . . . 5
bits ..^
|
| 138 | 137 | intnand 943 |
. . . 4
bits ..^
|
| 139 | simpll 531 |
. . . . 5
bits ..^
| |
| 140 | elznn0nn 9663 |
. . . . 5
 | |
| 141 | 139, 140 | sylib 122 |
. . . 4
bits ..^
|
| 142 | 138, 141 | ecased 1390 |
. . 3
bits ..^
|
| 143 | simplr 533 |
. . 3
bits ..^
| |
| 144 | simpr 110 |
. . 3
bits ..^
bits
..^ | |
| 145 | eqid 2238 |
. . 3
inf 
inf
| |
| 146 | 142, 143, 144, 145 | bitsfzolem 12739 |
. 2
bits ..^
..^ |
| 147 | 67, 146 | impbida 604 |
1
..^
bits
..^ |
| Colors of variables: wff set class |
| This proof depends on syntax axioms:
wn 3
wi 4
wa 104
wb 105 wo 720 DECID wdc 846
w3a 1009
wceq 1402
wcel 2209
crab 2532
wss 3220
cpr 3710
class class class wbr 4130 cfv 5377 (class class class)co 6085
csup 7323
infcinf 7324 cr 8179 cc0 8180 c1 8181 caddc 8183 cmul 8185 clt 8361 cle 8362 cneg 8500 cdiv 9005 cn 9307 c2 9358 cn0 9568 cz 9649 cuz 9931 cq 10029 crp 10065 ..^cfzo 10560
cfl 10714
cexp 10989
cdvds 12572 bitscbits 12725 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300 |
| This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-sup 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-n0 9569 df-z 9650 df-uz 9932 df-q 10030 df-rp 10066 df-fz 10423 df-fzo 10561 df-fl 10716 df-seqfrec 10899 df-exp 10990 df-cj 11622 df-re 11623 df-im 11624 df-rsqrt 11779 df-abs 11780 df-dvds 12573 df-bits 12726 |
| This theorem is used by: bitsfi 12742 0bits 12744 bitsinv1 12747 |
| Copyright terms: Public domain | W3C validator |