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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 12108 |
. . . 4
bits
    | |
| 2 | simp32 1036 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9636 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2289 |
. . . . . 6
..^
|
| 5 | simp1r 1024 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9446 |
. . . . . 6
..^
|
| 7 | 2re 9060 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 10782 |
. . . . . . . 8
..^
|
| 10 | simp1l 1023 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9448 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 10782 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8055 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8044 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9352 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 9700 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9152 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 10787 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 9698 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 411 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10336 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1037 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10333 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 411 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9104 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4041 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 9733 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9446 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 10789 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10231 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1021 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 9812 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9337 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10380 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 413 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12076 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4071 |
. . . . . . . . . . . . . . . 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 664 |
. . . . . . . . . . 11
..^
|
| 49 | notnotrdc 844 |
. . . . . . . . . . 11
DECID
| |
| 50 | 25, 48, 49 | sylc 62 |
. . . . . . . . . 10
..^
|
| 51 | 1red 8041 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 9821 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4057 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10232 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1021 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8151 |
. . . . . . 7
..^
|
| 58 | 1lt2 9160 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 10801 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1252 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10225 |
. . . . . 6
..^
| |
| 64 | 4, 6, 62, 63 | syl3anbrc 1183 |
. . . . 5
..^
..^ |
| 65 | 64 | 3expia 1207 |
. . . 4
..^
..^ |
| 66 | 1, 65 | biimtrid 152 |
. . 3
..^ bits ..^ |
| 67 | 66 | ssrdv 3189 |
. 2
..^ bits
..^ |
| 68 | simpr 110 |
. . . . . . . 8
bits ..^  | |
| 69 | 68 | nnred 9003 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 534 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9303 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11377 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 411 |
. . . . . 6
bits ..^ |
| 74 | simplr 528 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12078 |
. . . . . . . . . . 11
| |
| 76 | simplll 533 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9448 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9302 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11390 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 10787 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9040 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8041 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9449 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9004 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9036 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 8830 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9303 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9003 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11376 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9354 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9615 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 10754 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 414 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8145 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8150 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8043 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4061 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 9769 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 9825 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4055 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8554 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9034 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9034 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 8962 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4061 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8542 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 7972 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9095 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9101 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8171 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4075 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 9698 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 411 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9358 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10380 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 413 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 946 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4045 |
. . . . . . . . . . 11
bits ..^
|
| 125 | 75, 124 | mtbiri 676 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12109 |
. . . . . . . . . . 11
bits
| |
| 127 | 76, 81, 126 | syl2anc 411 |
. . . . . . . . . 10
bits ..^
bits
|
| 128 | 125, 127 | mpbird 167 |
. . . . . . . . 9
bits ..^
bits |
| 129 | 74, 128 | sseldd 3184 |
. . . . . . . 8
bits ..^
..^ |
| 130 | elfzolt2 10232 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9446 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9446 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9372 |
. . . . . . . 8
| |
| 135 | 132, 133, 134 | syl2anc 411 |
. . . . . . 7
bits ..^
|
| 136 | 131, 135 | mpbid 147 |
. . . . . 6
bits ..^
|
| 137 | 73, 136 | pm2.65da 662 |
. . . . 5
bits ..^
|
| 138 | 137 | intnand 932 |
. . . 4
bits ..^
|
| 139 | simpll 527 |
. . . . 5
bits ..^
| |
| 140 | elznn0nn 9340 |
. . . . 5
 | |
| 141 | 139, 140 | sylib 122 |
. . . 4
bits ..^
|
| 142 | 138, 141 | ecased 1360 |
. . 3
bits ..^
|
| 143 | simplr 528 |
. . 3
bits ..^
| |
| 144 | simpr 110 |
. . 3
bits ..^
bits
..^ | |
| 145 | eqid 2196 |
. . 3
inf 
inf
| |
| 146 | 142, 143, 144, 145 | bitsfzolem 12118 |
. 2
bits ..^
..^ |
| 147 | 67, 146 | impbida 596 |
1
..^
bits
..^ |
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 104
wb 105 wo 709 DECID wdc 835
w3a 980
wceq 1364
wcel 2167
crab 2479
wss 3157
cpr 3623
class class class wbr 4033 cfv 5258 (class class class)co 5922
csup 7048
infcinf 7049 cr 7878 cc0 7879 c1 7880 caddc 7882 cmul 7884 clt 8061 cle 8062 cneg 8198 cdiv 8699 cn 8990 c2 9041 cn0 9249 cz 9326 cuz 9601 cq 9693 crp 9728 ..^cfzo 10217
cfl 10358
cexp 10630
cdvds 11952 bitscbits 12105 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-bits 12106 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |