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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 12509 |
. . . 4
bits
    | |
| 2 | simp32 1060 |
. . . . . . 7
..^
| |
| 3 | nn0uz 9791 |
. . . . . . 7
  | |
| 4 | 2, 3 | eleqtrdi 2324 |
. . . . . 6
..^
|
| 5 | simp1r 1048 |
. . . . . . 7
..^
| |
| 6 | 5 | nn0zd 9600 |
. . . . . 6
..^
|
| 7 | 2re 9213 |
. . . . . . . . . 10
 | |
| 8 | 7 | a1i 9 |
. . . . . . . . 9
..^
|
| 9 | 8, 2 | reexpcld 10953 |
. . . . . . . 8
..^
|
| 10 | simp1l 1047 |
. . . . . . . . 9
..^
| |
| 11 | 10 | zred 9602 |
. . . . . . . 8
..^
|
| 12 | 8, 5 | reexpcld 10953 |
. . . . . . . 8
..^
|
| 13 | 9 | recnd 8208 |
. . . . . . . . . 10
..^
 |
| 14 | 13 | mullidd 8197 |
. . . . . . . . 9
..^
  |
| 15 | 1z 9505 |
. . . . . . . . . . . . . 14
| |
| 16 | zq 9860 |
. . . . . . . . . . . . . 14
 | |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 18 | 17 | a1i 9 |
. . . . . . . . . . . 12
..^
|
| 19 | 2nn 9305 |
. . . . . . . . . . . . . . 15
 | |
| 20 | 19 | a1i 9 |
. . . . . . . . . . . . . 14
..^
|
| 21 | 20, 2 | nnexpcld 10958 |
. . . . . . . . . . . . 13
..^
|
| 22 | znq 9858 |
. . . . . . . . . . . . 13
| |
| 23 | 10, 21, 22 | syl2anc 411 |
. . . . . . . . . . . 12
..^
|
| 24 | qdcle 10507 |
. . . . . . . . . . . 12
DECID 
| |
| 25 | 18, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
..^
DECID
|
| 26 | simp33 1061 |
. . . . . . . . . . . 12
..^
| |
| 27 | qltnle 10504 |
. . . . . . . . . . . . . 14
| |
| 28 | 23, 18, 27 | syl2anc 411 |
. . . . . . . . . . . . 13
..^
|
| 29 | 0p1e1 9257 |
. . . . . . . . . . . . . . 15
 | |
| 30 | 29 | breq2i 4096 |
. . . . . . . . . . . . . 14
|
| 31 | 2rp 9893 |
. . . . . . . . . . . . . . . . . 18
 | |
| 32 | 31 | a1i 9 |
. . . . . . . . . . . . . . . . 17
..^
|
| 33 | 2 | nn0zd 9600 |
. . . . . . . . . . . . . . . . 17
..^
|
| 34 | 32, 33 | rpexpcld 10960 |
. . . . . . . . . . . . . . . 16
..^
|
| 35 | elfzole1 10391 |
. . . . . . . . . . . . . . . . 17
..^
| |
| 36 | 35 | 3ad2ant2 1045 |
. . . . . . . . . . . . . . . 16
..^
|
| 37 | 11, 34, 36 | divge0d 9972 |
. . . . . . . . . . . . . . 15
..^
|
| 38 | 0z 9490 |
. . . . . . . . . . . . . . . . 17
| |
| 39 | flqbi 10551 |
. . . . . . . . . . . . . . . . 17
| |
| 40 | 23, 38, 39 | sylancl 413 |
. . . . . . . . . . . . . . . 16
..^
|
| 41 | z0even 12477 |
. . . . . . . . . . . . . . . . 17
| |
| 42 | id 19 |
. . . . . . . . . . . . . . . . 17
| |
| 43 | 41, 42 | breqtrrid 4126 |
. . . . . . . . . . . . . . . 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 850 |
. . . . . . . . . . 11
DECID
| |
| 50 | 25, 48, 49 | sylc 62 |
. . . . . . . . . 10
..^
|
| 51 | 1red 8194 |
. . . . . . . . . . 11
..^
| |
| 52 | 51, 11, 34 | lemuldivd 9981 |
. . . . . . . . . 10
..^
|
| 53 | 50, 52 | mpbird 167 |
. . . . . . . . 9
..^
|
| 54 | 14, 53 | eqbrtrrd 4112 |
. . . . . . . 8
..^
|
| 55 | elfzolt2 10392 |
. . . . . . . . 9
..^
| |
| 56 | 55 | 3ad2ant2 1045 |
. . . . . . . 8
..^
|
| 57 | 9, 11, 12, 54, 56 | lelttrd 8304 |
. . . . . . 7
..^
|
| 58 | 1lt2 9313 |
. . . . . . . . 9
| |
| 59 | 58 | a1i 9 |
. . . . . . . 8
..^
|
| 60 | nn0ltexp2 10972 |
. . . . . . . 8
| |
| 61 | 8, 2, 5, 59, 60 | syl31anc 1276 |
. . . . . . 7
..^
|
| 62 | 57, 61 | mpbird 167 |
. . . . . 6
..^
|
| 63 | elfzo2 10385 |
. . . . . 6
..^
| |
| 64 | 4, 6, 62, 63 | syl3anbrc 1207 |
. . . . 5
..^
..^ |
| 65 | 64 | 3expia 1231 |
. . . 4
..^
..^ |
| 66 | 1, 65 | biimtrid 152 |
. . 3
..^ bits ..^ |
| 67 | 66 | ssrdv 3233 |
. 2
..^ bits
..^ |
| 68 | simpr 110 |
. . . . . . . 8
bits ..^  | |
| 69 | 68 | nnred 9156 |
. . . . . . 7
bits ..^ |
| 70 | simpllr 536 |
. . . . . . . 8
bits ..^ | |
| 71 | 70 | nn0red 9456 |
. . . . . . 7
bits ..^ |
| 72 | maxle2 11777 |
. . . . . . 7
    | |
| 73 | 69, 71, 72 | syl2anc 411 |
. . . . . 6
bits ..^ |
| 74 | simplr 529 |
. . . . . . . . 9
bits ..^ bits
..^ | |
| 75 | n2dvdsm1 12479 |
. . . . . . . . . . 11
| |
| 76 | simplll 535 |
. . . . . . . . . . . . . . . 16
bits ..^ | |
| 77 | 76 | zred 9602 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 78 | 19 | a1i 9 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 79 | 68 | nnnn0d 9455 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 80 | nn0maxcl 11790 |
. . . . . . . . . . . . . . . . 17
| |
| 81 | 79, 70, 80 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 82 | 78, 81 | nnexpcld 10958 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 83 | 77, 82 | nndivred 9193 |
. . . . . . . . . . . . . 14
bits ..^ |
| 84 | 1red 8194 |
. . . . . . . . . . . . . 14
bits ..^ | |
| 85 | 76 | zcnd 9603 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 86 | 82 | nncnd 9157 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 87 | 82 | nnap0d 9189 |
. . . . . . . . . . . . . . . 16
bits ..^
# |
| 88 | 85, 86, 87 | divnegapd 8983 |
. . . . . . . . . . . . . . 15
bits ..^ |
| 89 | 81 | nn0red 9456 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 90 | 82 | nnred 9156 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 91 | maxle1 11776 |
. . . . . . . . . . . . . . . . . . 19
| |
| 92 | 69, 71, 91 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
bits ..^ |
| 93 | 2z 9507 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 94 | uzid 9770 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 95 | 93, 94 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
|
| 96 | bernneq3 10925 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 97 | 95, 81, 96 | sylancr 414 |
. . . . . . . . . . . . . . . . . . 19
bits ..^
|
| 98 | 89, 90, 97 | ltled 8298 |
. . . . . . . . . . . . . . . . . 18
bits ..^
|
| 99 | 69, 89, 90, 92, 98 | letrd 8303 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 100 | 86 | mulridd 8196 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 101 | 99, 100 | breqtrrd 4116 |
. . . . . . . . . . . . . . . 16
bits ..^ |
| 102 | 82 | nnrpd 9929 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 103 | 69, 84, 102 | ledivmuld 9985 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 104 | 101, 103 | mpbird 167 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 105 | 88, 104 | eqbrtrd 4110 |
. . . . . . . . . . . . . 14
bits ..^ |
| 106 | 83, 84, 105 | lenegcon1d 8707 |
. . . . . . . . . . . . 13
bits ..^
|
| 107 | 68 | nngt0d 9187 |
. . . . . . . . . . . . . . . . 17
bits ..^ |
| 108 | 82 | nngt0d 9187 |
. . . . . . . . . . . . . . . . 17
bits ..^
|
| 109 | 69, 90, 107, 108 | divgt0d 9115 |
. . . . . . . . . . . . . . . 16
bits ..^
|
| 110 | 109, 88 | breqtrrd 4116 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 111 | 83 | lt0neg1d 8695 |
. . . . . . . . . . . . . . 15
bits ..^
|
| 112 | 110, 111 | mpbird 167 |
. . . . . . . . . . . . . 14
bits ..^ |
| 113 | ax-1cn 8125 |
. . . . . . . . . . . . . . 15
| |
| 114 | neg1cn 9248 |
. . . . . . . . . . . . . . 15
| |
| 115 | 1pneg1e0 9254 |
. . . . . . . . . . . . . . 15
| |
| 116 | 113, 114, 115 | addcomli 8324 |
. . . . . . . . . . . . . 14
|
| 117 | 112, 116 | breqtrrdi 4130 |
. . . . . . . . . . . . 13
bits ..^ |
| 118 | znq 9858 |
. . . . . . . . . . . . . . 15
| |
| 119 | 76, 82, 118 | syl2anc 411 |
. . . . . . . . . . . . . 14
bits ..^ |
| 120 | neg1z 9511 |
. . . . . . . . . . . . . 14
| |
| 121 | flqbi 10551 |
. . . . . . . . . . . . . 14
| |
| 122 | 119, 120, 121 | sylancl 413 |
. . . . . . . . . . . . 13
bits ..^
|
| 123 | 106, 117, 122 | mpbir2and 952 |
. . . . . . . . . . . 12
bits ..^
|
| 124 | 123 | breq2d 4100 |
. . . . . . . . . . 11
bits ..^
|
| 125 | 75, 124 | mtbiri 681 |
. . . . . . . . . 10
bits ..^
|
| 126 | bitsval2 12510 |
. . . . . . . . . . 11
bits
| |
| 127 | 76, 81, 126 | syl2anc 411 |
. . . . . . . . . 10
bits ..^
bits
|
| 128 | 125, 127 | mpbird 167 |
. . . . . . . . 9
bits ..^
bits |
| 129 | 74, 128 | sseldd 3228 |
. . . . . . . 8
bits ..^
..^ |
| 130 | elfzolt2 10392 |
. . . . . . . 8
..^ | |
| 131 | 129, 130 | syl 14 |
. . . . . . 7
bits ..^
|
| 132 | 81 | nn0zd 9600 |
. . . . . . . 8
bits ..^
|
| 133 | 70 | nn0zd 9600 |
. . . . . . . 8
bits ..^ |
| 134 | zltnle 9525 |
. . . . . . . 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 938 |
. . . 4
bits ..^
|
| 139 | simpll 527 |
. . . . 5
bits ..^
| |
| 140 | elznn0nn 9493 |
. . . . 5
 | |
| 141 | 139, 140 | sylib 122 |
. . . 4
bits ..^
|
| 142 | 138, 141 | ecased 1385 |
. . 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 12520 |
. 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 715 DECID wdc 841
w3a 1004
wceq 1397
wcel 2202
crab 2514
wss 3200
cpr 3670
class class class wbr 4088 cfv 5326 (class class class)co 6018
csup 7181
infcinf 7182 cr 8031 cc0 8032 c1 8033 caddc 8035 cmul 8037 clt 8214 cle 8215 cneg 8351 cdiv 8852 cn 9143 c2 9194 cn0 9402 cz 9479 cuz 9755 cq 9853 crp 9888 ..^cfzo 10377
cfl 10529
cexp 10801
cdvds 12353 bitscbits 12506 |
| 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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 |
| This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-xor 1420 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-fz 10244 df-fzo 10378 df-fl 10531 df-seqfrec 10711 df-exp 10802 df-cj 11407 df-re 11408 df-im 11409 df-rsqrt 11563 df-abs 11564 df-dvds 12354 df-bits 12507 |
| This theorem is referenced by: bitsfi 12523 0bits 12525 bitsinv1 12528 |
| Copyright terms: Public domain | W3C validator |