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Theorem bitsuz 12941
Description: The bits of a number are all at least  N iff the number is divisible by  2 ^ N. (Contributed by Mario Carneiro, 21-Sep-2016.)
Assertion
Ref Expression
bitsuz  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( 2 ^ N )  ||  A  <->  (bits `  A )  C_  ( ZZ>=
`  N ) ) )

Proof of Theorem bitsuz
StepHypRef Expression
1 bitsres 12940 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )
21eqeq1d 2412 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  A )  i^i  ( ZZ>=
`  N ) )  =  (bits `  A
)  <->  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )  =  (bits `  A )
) )
3 simpl 444 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
43zred 10331 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  RR )
5 2nn 10089 . . . . . . . . 9  |-  2  e.  NN
65a1i 11 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
2  e.  NN )
7 simpr 448 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  NN0 )
86, 7nnexpcld 11499 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  NN )
94, 8nndivred 10004 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  /  (
2 ^ N ) )  e.  RR )
109flcld 11162 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( |_ `  ( A  /  ( 2 ^ N ) ) )  e.  ZZ )
118nnzd 10330 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  ZZ )
1210, 11zmulcld 10337 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  e.  ZZ )
13 bitsf1 12913 . . . . 5  |- bits : ZZ -1-1-> ~P
NN0
14 f1fveq 5967 . . . . 5  |-  ( (bits
: ZZ -1-1-> ~P NN0  /\  ( ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  e.  ZZ  /\  A  e.  ZZ ) )  ->  ( (bits `  ( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) ) )  =  (bits `  A )  <->  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  =  A ) )
1513, 14mpan 652 . . . 4  |-  ( ( ( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  e.  ZZ  /\  A  e.  ZZ )  ->  ( (bits `  (
( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )  =  (bits `  A
)  <->  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  =  A ) )
1612, 3, 15syl2anc 643 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  (
( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )  =  (bits `  A
)  <->  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  =  A ) )
17 dvdsmul2 12827 . . . . . 6  |-  ( ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  e.  ZZ  /\  (
2 ^ N )  e.  ZZ )  -> 
( 2 ^ N
)  ||  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
1810, 11, 17syl2anc 643 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  ||  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
19 breq2 4176 . . . . 5  |-  ( ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) )  =  A  ->  ( (
2 ^ N ) 
||  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  <->  ( 2 ^ N )  ||  A ) )
2018, 19syl5ibcom 212 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  =  A  ->  ( 2 ^ N )  ||  A
) )
218nnne0d 10000 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  =/=  0 )
22 dvdsval2 12810 . . . . . . . . . 10  |-  ( ( ( 2 ^ N
)  e.  ZZ  /\  ( 2 ^ N
)  =/=  0  /\  A  e.  ZZ )  ->  ( ( 2 ^ N )  ||  A 
<->  ( A  /  (
2 ^ N ) )  e.  ZZ ) )
2311, 21, 3, 22syl3anc 1184 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( 2 ^ N )  ||  A  <->  ( A  /  ( 2 ^ N ) )  e.  ZZ ) )
2423biimpa 471 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( A  /  (
2 ^ N ) )  e.  ZZ )
25 flid 11171 . . . . . . . 8  |-  ( ( A  /  ( 2 ^ N ) )  e.  ZZ  ->  ( |_ `  ( A  / 
( 2 ^ N
) ) )  =  ( A  /  (
2 ^ N ) ) )
2624, 25syl 16 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( |_ `  ( A  /  ( 2 ^ N ) ) )  =  ( A  / 
( 2 ^ N
) ) )
2726oveq1d 6055 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  =  ( ( A  /  ( 2 ^ N ) )  x.  ( 2 ^ N ) ) )
283zcnd 10332 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
2928adantr 452 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  A  e.  CC )
308nncnd 9972 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  CC )
3130adantr 452 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( 2 ^ N
)  e.  CC )
32 2cn 10026 . . . . . . . . 9  |-  2  e.  CC
3332a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  2  e.  CC )
34 2ne0 10039 . . . . . . . . 9  |-  2  =/=  0
3534a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  2  =/=  0 )
367nn0zd 10329 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ZZ )
3736adantr 452 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  N  e.  ZZ )
3833, 35, 37expne0d 11484 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( 2 ^ N
)  =/=  0 )
3929, 31, 38divcan1d 9747 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( ( A  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  A )
4027, 39eqtrd 2436 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  =  A )
4140ex 424 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( 2 ^ N )  ||  A  ->  ( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  =  A ) )
4220, 41impbid 184 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  =  A  <-> 
( 2 ^ N
)  ||  A )
)
432, 16, 423bitrrd 272 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( 2 ^ N )  ||  A  <->  ( (bits `  A )  i^i  ( ZZ>= `  N )
)  =  (bits `  A ) ) )
44 df-ss 3294 . 2  |-  ( (bits `  A )  C_  ( ZZ>=
`  N )  <->  ( (bits `  A )  i^i  ( ZZ>=
`  N ) )  =  (bits `  A
) )
4543, 44syl6bbr 255 1  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( 2 ^ N )  ||  A  <->  (bits `  A )  C_  ( ZZ>=
`  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   class class class wbr 4172   -1-1->wf1 5410   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    x. cmul 8951    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   |_cfl 11156   ^cexp 11337    || cdivides 12807  bitscbits 12886
This theorem is referenced by:  bitsshft  12942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-had 1386  df-cad 1387  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808  df-bits 12889  df-sad 12918
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