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| Mirrors > Home > MPE Home > Th. List > bitsfzo | Structured version Visualization version Unicode version | ||
Description: The bits of a number are
all 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 15146 |
. . . 4
bits
    | |
| 2 | simp32 1098 |
. . . . . . 7
..^
| |
| 3 | nn0uz 11722 |
. . . . . . 7
  | |
| 4 | 2, 3 | syl6eleq 2711 |
. . . . . 6
..^
|
| 5 | simp1r 1086 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 11480 |
. . . . . 6
..^
|
| 7 | 2re 11090 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 11 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 13025 |
. . . . . . . 8
..^
|
| 10 | simp1l 1085 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 11482 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 13025 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 10068 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mulid2d 10058 |
. . . . . . . . 9
..^
  |
| 15 | simp33 1099 |
. . . . . . . . . . 11
..^
| |
| 16 | 2rp 11837 |
. . . . . . . . . . . . . . . 16
 | |
| 17 | 16 | a1i 11 |
. . . . . . . . . . . . . . 15
..^
|
| 18 | 2 | nn0zd 11480 |
. . . . . . . . . . . . . . 15
..^
|
| 19 | 17, 18 | rpexpcld 13032 |
. . . . . . . . . . . . . 14
..^
|
| 20 | 11, 19 | rerpdivcld 11903 |
. . . . . . . . . . . . 13
..^
|
| 21 | 1red 10055 |
. . . . . . . . . . . . 13
..^
| |
| 22 | 20, 21 | ltnled 10184 |
. . . . . . . . . . . 12
..^
 |
| 23 | 0p1e1 11132 |
. . . . . . . . . . . . . 14
 | |
| 24 | 23 | breq2i 4661 |
. . . . . . . . . . . . 13
|
| 25 | elfzole1 12478 |
. . . . . . . . . . . . . . . 16
..^
| |
| 26 | 25 | 3ad2ant2 1083 |
. . . . . . . . . . . . . . 15
..^
|
| 27 | 11, 19, 26 | divge0d 11912 |
. . . . . . . . . . . . . 14
..^
|
| 28 | 0z 11388 |
. . . . . . . . . . . . . . . 16
| |
| 29 | flbi 12617 |
. . . . . . . . . . . . . . . 16
| |
| 30 | 20, 28, 29 | sylancl 694 |
. . . . . . . . . . . . . . 15
..^
|
| 31 | 2z 11409 |
. . . . . . . . . . . . . . . . 17
| |
| 32 | dvds0 14997 |
. . . . . . . . . . . . . . . . 17
| |
| 33 | 31, 32 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
|
| 34 | id 22 |
. . . . . . . . . . . . . . . 16
| |
| 35 | 33, 34 | syl5breqr 4691 |
. . . . . . . . . . . . . . 15
|
| 36 | 30, 35 | syl6bir 244 |
. . . . . . . . . . . . . 14
..^
|
| 37 | 27, 36 | mpand 711 |
. . . . . . . . . . . . 13
..^
|
| 38 | 24, 37 | syl5bir 233 |
. . . . . . . . . . . 12
..^
|
| 39 | 22, 38 | sylbird 250 |
. . . . . . . . . . 11
..^
|
| 40 | 15, 39 | mt3d 140 |
. . . . . . . . . 10
..^
|
| 41 | 21, 11, 19 | lemuldivd 11921 |
. . . . . . . . . 10
..^
|
| 42 | 40, 41 | mpbird 247 |
. . . . . . . . 9
..^
|
| 43 | 14, 42 | eqbrtrrd 4677 |
. . . . . . . 8
..^
|
| 44 | elfzolt2 12479 |
. . . . . . . . 9
..^
| |
| 45 | 44 | 3ad2ant2 1083 |
. . . . . . . 8
..^
|
| 46 | 9, 11, 12, 43, 45 | lelttrd 10195 |
. . . . . . 7
..^
|
| 47 | 1lt2 11194 |
. . . . . . . . 9
| |
| 48 | 47 | a1i 11 |
. . . . . . . 8
..^
|
| 49 | 8, 18, 6, 48 | ltexp2d 13038 |
. . . . . . 7
..^
|
| 50 | 46, 49 | mpbird 247 |
. . . . . 6
..^
|
| 51 | elfzo2 12473 |
. . . . . 6
..^
| |
| 52 | 4, 6, 50, 51 | syl3anbrc 1246 |
. . . . 5
..^
..^ |
| 53 | 52 | 3expia 1267 |
. . . 4
..^
..^ |
| 54 | 1, 53 | syl5bi 232 |
. . 3
..^ bits ..^ |
| 55 | 54 | ssrdv 3609 |
. 2
..^ bits
..^ |
| 56 | simpr 477 |
. . . . . . . 8
bits ..^   | |
| 57 | 56 | nnred 11035 |
. . . . . . 7
bits ..^ |
| 58 | simpllr 799 |
. . . . . . . 8
bits ..^ | |
| 59 | 58 | nn0red 11352 |
. . . . . . 7
bits ..^ |
| 60 | max2 12018 |
. . . . . . 7
  | |
| 61 | 57, 59, 60 | syl2anc 693 |
. . . . . 6
bits ..^ |
| 62 | simplr 792 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 63 | n2dvds1 15104 |
. . . . . . . . . . . 12
| |
| 64 | 1z 11407 |
. . . . . . . . . . . . 13
| |
| 65 | dvdsnegb 14999 |
. . . . . . . . . . . . 13
| |
| 66 | 31, 64, 65 | mp2an 708 |
. . . . . . . . . . . 12
|
| 67 | 63, 66 | mtbi 312 |
. . . . . . . . . . 11
|
| 68 | simplll 798 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 69 | 68 | zred 11482 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 70 | 2nn 11185 |
. . . . . . . . . . . . . . . . 17
| |
| 71 | 70 | a1i 11 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 72 | 56 | nnnn0d 11351 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 73 | 58, 72 | ifcld 4131 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 74 | 71, 73 | nnexpcld 13030 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 75 | 69, 74 | nndivred 11069 |
. . . . . . . . . . . . . 14
bits ..^ |
| 76 | 1red 10055 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 77 | 68 | zcnd 11483 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 78 | 74 | nncnd 11036 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 2cnd 11093 |
. . . . . . . . . . . . . . . . 17
bits ..^ | |
| 80 | 2ne0 11113 |
. . . . . . . . . . . . . . . . . 18
 | |
| 81 | 80 | a1i 11 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 82 | 73 | nn0zd 11480 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 83 | 79, 81, 82 | expne0d 13014 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 84 | 77, 78, 83 | divnegd 10814 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 85 | 73 | nn0red 11352 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 86 | 74 | nnred 11035 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 87 | max1 12016 |
. . . . . . . . . . . . . . . . . . 19
| |
| 88 | 57, 59, 87 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 89 | uzid 11702 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 90 | 31, 89 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 91 | bernneq3 12992 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 92 | 90, 73, 91 | sylancr 695 |
. . . . . . . . . . . . . . . . . . 19
bits ..^ |
| 93 | 85, 86, 92 | ltled 10185 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 94 | 57, 85, 86, 88, 93 | letrd 10194 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 95 | 78 | mulid1d 10057 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 96 | 94, 95 | breqtrrd 4681 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 97 | 74 | nnrpd 11870 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 98 | 57, 76, 97 | ledivmuld 11925 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 99 | 96, 98 | mpbird 247 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 100 | 84, 99 | eqbrtrd 4675 |
. . . . . . . . . . . . . 14
bits ..^ |
| 101 | 75, 76, 100 | lenegcon1d 10609 |
. . . . . . . . . . . . 13
bits ..^ |
| 102 | 56 | nngt0d 11064 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 103 | 74 | nngt0d 11064 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 104 | 57, 86, 102, 103 | divgt0d 10959 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 105 | 104, 84 | breqtrrd 4681 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 106 | 75 | lt0neg1d 10597 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 107 | 105, 106 | mpbird 247 |
. . . . . . . . . . . . . 14
bits ..^ |
| 108 | ax-1cn 9994 |
. . . . . . . . . . . . . . 15
| |
| 109 | neg1cn 11124 |
. . . . . . . . . . . . . . 15
| |
| 110 | 1pneg1e0 11129 |
. . . . . . . . . . . . . . 15
| |
| 111 | 108, 109, 110 | addcomli 10228 |
. . . . . . . . . . . . . 14
|
| 112 | 107, 111 | syl6breqr 4695 |
. . . . . . . . . . . . 13
bits ..^ |
| 113 | neg1z 11413 |
. . . . . . . . . . . . . 14
| |
| 114 | flbi 12617 |
. . . . . . . . . . . . . 14
| |
| 115 | 75, 113, 114 | sylancl 694 |
. . . . . . . . . . . . 13
bits ..^
|
| 116 | 101, 112, 115 | mpbir2and 957 |
. . . . . . . . . . . 12
bits ..^
|
| 117 | 116 | breq2d 4665 |
. . . . . . . . . . 11
bits ..^ |
| 118 | 67, 117 | mtbiri 317 |
. . . . . . . . . 10
bits ..^
|
| 119 | bitsval2 15147 |
. . . . . . . . . . 11
bits
| |
| 120 | 68, 73, 119 | syl2anc 693 |
. . . . . . . . . 10
bits ..^ bits
|
| 121 | 118, 120 | mpbird 247 |
. . . . . . . . 9
bits ..^ bits |
| 122 | 62, 121 | sseldd 3604 |
. . . . . . . 8
bits ..^ ..^ |
| 123 | elfzolt2 12479 |
. . . . . . . 8
..^
| |
| 124 | 122, 123 | syl 17 |
. . . . . . 7
bits ..^ |
| 125 | 85, 59 | ltnled 10184 |
. . . . . . 7
bits ..^
|
| 126 | 124, 125 | mpbid 222 |
. . . . . 6
bits ..^
|
| 127 | 61, 126 | pm2.65da 600 |
. . . . 5
bits ..^
|
| 128 | 127 | intnand 962 |
. . . 4
bits ..^
|
| 129 | simpll 790 |
. . . . . 6
bits ..^
| |
| 130 | elznn0nn 11391 |
. . . . . 6
 | |
| 131 | 129, 130 | sylib 208 |
. . . . 5
bits ..^
|
| 132 | 131 | ord 392 |
. . . 4
bits ..^
|
| 133 | 128, 132 | mt3d 140 |
. . 3
bits ..^
|
| 134 | simplr 792 |
. . 3
bits ..^
| |
| 135 | simpr 477 |
. . 3
bits ..^
bits
..^ | |
| 136 | eqid 2622 |
. . 3
inf  
inf
| |
| 137 | 133, 134, 135, 136 | bitsfzolem 15156 |
. 2
bits ..^
..^ |
| 138 | 55, 137 | impbida 877 |
1
..^
bits
..^ |
| Colors of variables: wff setvar class |
| Syntax hints: wn 3
wi 4
wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 crab 2916 wss 3574 cif 4086 class class class wbr 4653
cfv 5888
(class class class)co 6650 infcinf 8347
cr 9935
cc0 9936
c1 9937
caddc 9939 cmul 9941 clt 10074 cle 10075 cneg 10267 cdiv 10684 cn 11020 c2 11070 cn0 11292 cz 11377 cuz 11687 crp 11832 ..^cfzo 12465
cfl 12591
cexp 12860
cdvds 14983 bitscbits 15141 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-dvds 14984 df-bits 15144 |
| This theorem is referenced by: bitsfi 15159 0bits 15161 bitsinv1 15164 sadcaddlem 15179 sadaddlem 15188 sadasslem 15192 sadeq 15194 |
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