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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 12565 |
. . . 4
bits
    | |
| 2 | simp32 1061 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9834 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2324 |
. . . . . 6
..^
|
| 5 | simp1r 1049 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9643 |
. . . . . 6
..^
|
| 7 | 2re 9256 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 10996 |
. . . . . . . 8
..^
|
| 10 | simp1l 1048 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9645 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 10996 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8251 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8240 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9548 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 9903 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9348 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 11001 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 9901 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 411 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10550 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1062 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10547 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 411 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9300 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4101 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 9936 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9643 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 11003 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10434 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1046 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 10015 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9533 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10594 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 413 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12533 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4131 |
. . . . . . . . . . . . . . . 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 669 |
. . . . . . . . . . 11
..^
|
| 49 | notnotrdc 851 |
. . . . . . . . . . 11
DECID
| |
| 50 | 25, 48, 49 | sylc 62 |
. . . . . . . . . 10
..^
|
| 51 | 1red 8237 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 10024 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4117 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10435 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1046 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8347 |
. . . . . . 7
..^
|
| 58 | 1lt2 9356 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 11015 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1277 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10428 |
. . . . . 6
..^
| |
| 64 | 4, 6, 62, 63 | syl3anbrc 1208 |
. . . . 5
..^
..^ |
| 65 | 64 | 3expia 1232 |
. . . 4
..^
..^ |
| 66 | 1, 65 | biimtrid 152 |
. . 3
..^ bits ..^ |
| 67 | 66 | ssrdv 3234 |
. 2
..^ bits
..^ |
| 68 | simpr 110 |
. . . . . . . 8
bits ..^  | |
| 69 | 68 | nnred 9199 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 536 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9499 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11833 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 411 |
. . . . . 6
bits ..^ |
| 74 | simplr 529 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12535 |
. . . . . . . . . . 11
| |
| 76 | simplll 535 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9645 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9498 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11846 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 11001 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9236 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8237 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9646 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9200 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9232 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 9026 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9499 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9199 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11832 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9550 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9813 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 10968 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 414 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8341 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8346 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8239 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4121 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 9972 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 10028 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4115 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8750 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9230 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9230 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9158 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4121 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8738 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 8168 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9291 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9297 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8367 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4135 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 9901 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 411 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9554 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10594 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 413 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 953 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4105 |
. . . . . . . . . . 11
bits ..^
|
| 125 | 75, 124 | mtbiri 682 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12566 |
. . . . . . . . . . 11
bits
| |
| 127 | 76, 81, 126 | syl2anc 411 |
. . . . . . . . . 10
bits ..^
bits
|
| 128 | 125, 127 | mpbird 167 |
. . . . . . . . 9
bits ..^
bits |
| 129 | 74, 128 | sseldd 3229 |
. . . . . . . 8
bits ..^
..^ |
| 130 | elfzolt2 10435 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9643 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9643 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9568 |
. . . . . . . 8
| |
| 135 | 132, 133, 134 | syl2anc 411 |
. . . . . . 7
bits ..^
|
| 136 | 131, 135 | mpbid 147 |
. . . . . 6
bits ..^
|
| 137 | 73, 136 | pm2.65da 667 |
. . . . 5
bits ..^
|
| 138 | 137 | intnand 939 |
. . . 4
bits ..^
|
| 139 | simpll 527 |
. . . . 5
bits ..^
| |
| 140 | elznn0nn 9536 |
. . . . 5
 | |
| 141 | 139, 140 | sylib 122 |
. . . 4
bits ..^
|
| 142 | 138, 141 | ecased 1386 |
. . 3
bits ..^
|
| 143 | simplr 529 |
. . 3
bits ..^
| |
| 144 | simpr 110 |
. . 3
bits ..^
bits
..^ | |
| 145 | eqid 2231 |
. . 3
inf 
inf
| |
| 146 | 142, 143, 144, 145 | bitsfzolem 12576 |
. 2
bits ..^
..^ |
| 147 | 67, 146 | impbida 600 |
1
..^
bits
..^ |
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 104
wb 105 wo 716 DECID wdc 842
w3a 1005
wceq 1398
wcel 2202
crab 2515
wss 3201
cpr 3674
class class class wbr 4093 cfv 5333 (class class class)co 6028
csup 7224
infcinf 7225 cr 8074 cc0 8075 c1 8076 caddc 8078 cmul 8080 clt 8257 cle 8258 cneg 8394 cdiv 8895 cn 9186 c2 9237 cn0 9445 cz 9522 cuz 9798 cq 9896 crp 9931 ..^cfzo 10420
cfl 10572
cexp 10844
cdvds 12409 bitscbits 12562 |
| 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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195 |
| This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-sup 7226 df-inf 7227 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-n0 9446 df-z 9523 df-uz 9799 df-q 9897 df-rp 9932 df-fz 10287 df-fzo 10421 df-fl 10574 df-seqfrec 10754 df-exp 10845 df-cj 11463 df-re 11464 df-im 11465 df-rsqrt 11619 df-abs 11620 df-dvds 12410 df-bits 12563 |
| This theorem is referenced by: bitsfi 12579 0bits 12581 bitsinv1 12584 |
| Copyright terms: Public domain | W3C validator |