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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 12712 |
. . . 4
bits
    | |
| 2 | simp32 1065 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9959 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2331 |
. . . . . 6
..^
|
| 5 | simp1r 1053 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9768 |
. . . . . 6
..^
|
| 7 | 2re 9375 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 11130 |
. . . . . . . 8
..^
|
| 10 | simp1l 1052 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9770 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 11130 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8354 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8344 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9672 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 10028 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9468 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 11135 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 10026 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 415 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10683 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 415 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1066 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10680 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 415 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9419 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4138 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 10061 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9768 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 11137 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10565 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1050 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 10140 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9657 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10727 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 417 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12680 |
. . . . . . . . . . . . . . . . 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 8341 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 10149 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4154 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10566 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1050 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8451 |
. . . . . . 7
..^
|
| 58 | 1lt2 9476 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 11149 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1281 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10559 |
. . . . . 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 9318 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 540 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9623 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11980 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 415 |
. . . . . 6
bits ..^ |
| 74 | simplr 533 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12682 |
. . . . . . . . . . 11
| |
| 76 | simplll 539 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9770 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9622 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11993 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 415 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 11135 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9355 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8341 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9771 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9319 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9351 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 9134 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9623 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9318 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11979 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 415 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9674 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9938 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 11102 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 418 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8445 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8450 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8343 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4158 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 10097 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 10153 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4152 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8855 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9349 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9349 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9266 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4158 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8843 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 8272 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9410 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9416 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8471 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4172 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 10026 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 415 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9678 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10727 |
. . . . . . . . . . . . . 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 12713 |
. . . . . . . . . . 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 10566 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9768 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9768 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9692 |
. . . . . . . 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 9660 |
. . . . 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 12723 |
. 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 7322
infcinf 7323 cr 8178 cc0 8179 c1 8180 caddc 8182 cmul 8184 clt 8360 cle 8361 cneg 8498 cdiv 9003 cn 9305 c2 9356 cn0 9565 cz 9646 cuz 9923 cq 10021 crp 10056 ..^cfzo 10551
cfl 10705
cexp 10977
cdvds 12556 bitscbits 12709 |
| 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 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299 |
| 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 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-n0 9566 df-z 9647 df-uz 9924 df-q 10022 df-rp 10057 df-fz 10414 df-fzo 10552 df-fl 10707 df-seqfrec 10887 df-exp 10978 df-cj 11609 df-re 11610 df-im 11611 df-rsqrt 11766 df-abs 11767 df-dvds 12557 df-bits 12710 |
| This theorem is used by: bitsfi 12726 0bits 12728 bitsinv1 12731 |
| Copyright terms: Public domain | W3C validator |