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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 12736 |
. . . 4
bits
    | |
| 2 | simp32 1065 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9974 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2331 |
. . . . . 6
..^
|
| 5 | simp1r 1053 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9778 |
. . . . . 6
..^
|
| 7 | 2re 9384 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 11150 |
. . . . . . . 8
..^
|
| 10 | simp1l 1052 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9780 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 11150 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8361 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8351 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9682 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 10043 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9478 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 11155 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 10041 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 415 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10699 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 415 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1066 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10696 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 415 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9428 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4139 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 10077 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9778 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 11157 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10581 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1050 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 10156 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9667 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10747 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 417 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12704 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4169 |
. . . . . . . . . . . . . . . 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 8348 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 10165 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4155 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10582 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1050 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8459 |
. . . . . . 7
..^
|
| 58 | 1lt2 9486 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 11170 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1281 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10575 |
. . . . . 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 9327 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 540 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9633 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 12002 |
. . . . . . 7
 
  | |
| 73 | 69, 71, 72 | syl2anc 415 |
. . . . . 6
bits ..^
|
| 74 | simplr 533 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12706 |
. . . . . . . . . . 11
| |
| 76 | simplll 539 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9780 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9632 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 12015 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 415 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 82 | 78, 81 | nnexpcld 11155 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 83 | 77, 82 | nndivred 9364 |
. . . . . . . . . . . . . 14
bits ..^
|
| 84 | 1red 8348 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9781 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9328 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 87 | 82 | nnap0d 9360 |
. . . . . . . . . . . . . . . 16
bits ..^ # |
| 88 | 85, 86, 87 | divnegapd 9142 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 89 | 81 | nn0red 9633 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 90 | 82 | nnred 9327 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 91 | maxle1 12001 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 415 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 93 | 2z 9684 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9953 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 11122 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 418 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8453 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8458 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8350 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4159 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 102 | 82 | nnrpd 10113 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 103 | 69, 84, 102 | ledivmuld 10169 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 105 | 88, 104 | eqbrtrd 4153 |
. . . . . . . . . . . . . 14
bits ..^
|
| 106 | 83, 84, 105 | lenegcon1d 8863 |
. . . . . . . . . . . . 13
bits ..^ |
| 107 | 68 | nngt0d 9358 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9358 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9274 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4159 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 111 | 83 | lt0neg1d 8851 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^
|
| 113 | ax-1cn 8279 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9419 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9425 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8479 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4173 |
. . . . . . . . . . . . 13
bits ..^
|
| 118 | znq 10041 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 415 |
. . . . . . . . . . . . . 14
bits ..^
|
| 120 | neg1z 9688 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10747 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 417 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 957 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4143 |
. . . . . . . . . . 11
bits ..^ |
| 125 | 75, 124 | mtbiri 686 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12737 |
. . . . . . . . . . 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 10582 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^ |
| 132 | 81 | nn0zd 9778 |
. . . . . . . 8
bits ..^ |
| 133 | 70 | nn0zd 9778 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9702 |
. . . . . . . 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 9670 |
. . . . 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 12747 |
. 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 4131 cfv 5380 (class class class)co 6088
csup 7329
infcinf 7330 cr 8185 cc0 8186 c1 8187 caddc 8189 cmul 8191 clt 8367 cle 8368 cneg 8506 cdiv 9011 cn 9314 c2 9365 cn0 9575 cz 9656 cuz 9938 cq 10036 crp 10072 ..^cfzo 10567
cfl 10721
cexp 10997
cdvds 12580 bitscbits 12733 |
| 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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306 |
| 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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-q 10037 df-rp 10073 df-fz 10430 df-fzo 10568 df-fl 10723 df-seqfrec 10907 df-exp 10998 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788 df-dvds 12581 df-bits 12734 |
| This theorem is used by: bitsfi 12750 0bits 12752 bitsinv1 12755 |
| Copyright terms: Public domain | W3C validator |