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Theorem bitsuz 12988
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 12987 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )
21eqeq1d 2446 . . 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 445 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
43zred 10377 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  RR )
5 2nn 10135 . . . . . . . . 9  |-  2  e.  NN
65a1i 11 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
2  e.  NN )
7 simpr 449 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  NN0 )
86, 7nnexpcld 11546 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  NN )
94, 8nndivred 10050 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  /  (
2 ^ N ) )  e.  RR )
109flcld 11209 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( |_ `  ( A  /  ( 2 ^ N ) ) )  e.  ZZ )
118nnzd 10376 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  ZZ )
1210, 11zmulcld 10383 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  e.  ZZ )
13 bitsf1 12960 . . . . 5  |- bits : ZZ -1-1-> ~P
NN0
14 f1fveq 6010 . . . . 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 653 . . . 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 644 . . 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 12874 . . . . . 6  |-  ( ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  e.  ZZ  /\  (
2 ^ N )  e.  ZZ )  -> 
( 2 ^ N
)  ||  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
1810, 11, 17syl2anc 644 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  ||  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
19 breq2 4218 . . . . 5  |-  ( ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) )  =  A  ->  ( (
2 ^ N ) 
||  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  <->  ( 2 ^ N )  ||  A ) )
2018, 19syl5ibcom 213 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  =  A  ->  ( 2 ^ N )  ||  A
) )
218nnne0d 10046 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  =/=  0 )
22 dvdsval2 12857 . . . . . . . . . 10  |-  ( ( ( 2 ^ N
)  e.  ZZ  /\  ( 2 ^ N
)  =/=  0  /\  A  e.  ZZ )  ->  ( ( 2 ^ N )  ||  A 
<->  ( A  /  (
2 ^ N ) )  e.  ZZ ) )
2311, 21, 3, 22syl3anc 1185 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( 2 ^ N )  ||  A  <->  ( A  /  ( 2 ^ N ) )  e.  ZZ ) )
2423biimpa 472 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( A  /  (
2 ^ N ) )  e.  ZZ )
25 flid 11218 . . . . . . . 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 6098 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  =  ( ( A  /  ( 2 ^ N ) )  x.  ( 2 ^ N ) ) )
283zcnd 10378 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
2928adantr 453 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  A  e.  CC )
308nncnd 10018 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  CC )
3130adantr 453 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( 2 ^ N
)  e.  CC )
32 2cn 10072 . . . . . . . . 9  |-  2  e.  CC
3332a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  2  e.  CC )
34 2ne0 10085 . . . . . . . . 9  |-  2  =/=  0
3534a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  2  =/=  0 )
367nn0zd 10375 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ZZ )
3736adantr 453 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  N  e.  ZZ )
3833, 35, 37expne0d 11531 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( 2 ^ N
)  =/=  0 )
3929, 31, 38divcan1d 9793 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( ( A  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  A )
4027, 39eqtrd 2470 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  N  e.  NN0 )  /\  ( 2 ^ N
)  ||  A )  ->  ( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  =  A )
4140ex 425 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( 2 ^ N )  ||  A  ->  ( ( |_ `  ( A  /  (
2 ^ N ) ) )  x.  (
2 ^ N ) )  =  A ) )
4220, 41impbid 185 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) )  =  A  <-> 
( 2 ^ N
)  ||  A )
)
432, 16, 423bitrrd 273 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( 2 ^ N )  ||  A  <->  ( (bits `  A )  i^i  ( ZZ>= `  N )
)  =  (bits `  A ) ) )
44 df-ss 3336 . 2  |-  ( (bits `  A )  C_  ( ZZ>=
`  N )  <->  ( (bits `  A )  i^i  ( ZZ>=
`  N ) )  =  (bits `  A
) )
4543, 44syl6bbr 256 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   class class class wbr 4214   -1-1->wf1 5453   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    x. cmul 8997    / cdiv 9679   NNcn 10002   2c2 10051   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   |_cfl 11203   ^cexp 11384    || cdivides 12854  bitscbits 12933
This theorem is referenced by:  bitsshft  12989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-xor 1315  df-tru 1329  df-had 1390  df-cad 1391  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-dvds 12855  df-bits 12936  df-sad 12965
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