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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 12693 |
. . . 4
bits
    | |
| 2 | simp32 1065 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9940 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2331 |
. . . . . 6
..^
|
| 5 | simp1r 1053 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9749 |
. . . . . 6
..^
|
| 7 | 2re 9357 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 11111 |
. . . . . . . 8
..^
|
| 10 | simp1l 1052 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9751 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 11111 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8348 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8338 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9653 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 10009 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9449 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 11116 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 10007 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 415 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10664 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 415 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1066 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10661 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 415 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9401 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4136 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 10042 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9749 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 11118 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10546 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1050 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 10121 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9638 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10708 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 417 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12661 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4166 |
. . . . . . . . . . . . . . . 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 8335 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 10130 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4152 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10547 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1050 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8445 |
. . . . . . 7
..^
|
| 58 | 1lt2 9457 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 11130 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1281 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10540 |
. . . . . 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 9300 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 540 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9604 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11961 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 415 |
. . . . . 6
bits ..^ |
| 74 | simplr 533 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12663 |
. . . . . . . . . . 11
| |
| 76 | simplll 539 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9751 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9603 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11974 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 415 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 11116 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9337 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8335 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9752 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9301 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9333 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 9127 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9604 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9300 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11960 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 415 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9655 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9919 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 11083 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 418 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8439 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8444 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8337 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4156 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 10078 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 10134 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4150 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8849 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9331 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9331 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9259 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4156 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8837 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 8266 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9392 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9398 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8465 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4170 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 10007 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 415 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9659 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10708 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 417 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 957 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4140 |
. . . . . . . . . . 11
bits ..^
|
| 125 | 75, 124 | mtbiri 686 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12694 |
. . . . . . . . . . 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 10547 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9749 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9749 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9673 |
. . . . . . . 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 9641 |
. . . . 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 12704 |
. 2
bits ..^
..^ |
| 147 | 67, 146 | impbida 604 |
1
..^
bits
..^ |
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 104
wb 105 wo 720 DECID wdc 846
w3a 1009
wceq 1402
wcel 2209
crab 2532
wss 3220
cpr 3709
class class class wbr 4128 cfv 5375 (class class class)co 6079
csup 7316
infcinf 7317 cr 8172 cc0 8173 c1 8174 caddc 8176 cmul 8178 clt 8354 cle 8355 cneg 8492 cdiv 8996 cn 9287 c2 9338 cn0 9546 cz 9627 cuz 9904 cq 10002 crp 10037 ..^cfzo 10532
cfl 10686
cexp 10958
cdvds 12537 bitscbits 12690 |
| 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 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 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292 ax-caucvg 8293 |
| This theorem 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 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-isom 5384 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-frec 6656 df-sup 7318 df-inf 7319 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-n0 9547 df-z 9628 df-uz 9905 df-q 10003 df-rp 10038 df-fz 10395 df-fzo 10533 df-fl 10688 df-seqfrec 10868 df-exp 10959 df-cj 11590 df-re 11591 df-im 11592 df-rsqrt 11747 df-abs 11748 df-dvds 12538 df-bits 12691 |
| This theorem is referenced by: bitsfi 12707 0bits 12709 bitsinv1 12712 |
| Copyright terms: Public domain | W3C validator |