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Theorem bitsp1 12943
Description: The  M  +  1-th bit of  N is the  M-th bit of  |_ ( N  /  2 ). (Contributed by Mario Carneiro, 5-Sep-2016.)
Assertion
Ref Expression
bitsp1  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( M  + 
1 )  e.  (bits `  N )  <->  M  e.  (bits `  ( |_ `  ( N  /  2
) ) ) ) )

Proof of Theorem bitsp1
StepHypRef Expression
1 2nn 10133 . . . . . . . . . . . 12  |-  2  e.  NN
21a1i 11 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
2  e.  NN )
32nncnd 10016 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
2  e.  CC )
4 simpr 448 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  M  e.  NN0 )
53, 4expp1d 11524 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ ( M  +  1 ) )  =  ( ( 2 ^ M )  x.  2 ) )
62, 4nnexpcld 11544 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ M
)  e.  NN )
76nncnd 10016 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ M
)  e.  CC )
87, 3mulcomd 9109 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 2 ^ M )  x.  2 )  =  ( 2  x.  ( 2 ^ M ) ) )
95, 8eqtrd 2468 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ ( M  +  1 ) )  =  ( 2  x.  ( 2 ^ M ) ) )
109oveq2d 6097 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( N  /  (
2 ^ ( M  +  1 ) ) )  =  ( N  /  ( 2  x.  ( 2 ^ M
) ) ) )
11 simpl 444 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  N  e.  ZZ )
1211zcnd 10376 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  N  e.  CC )
132nnne0d 10044 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
2  =/=  0 )
146nnne0d 10044 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ M
)  =/=  0 )
1512, 3, 7, 13, 14divdiv1d 9821 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( N  / 
2 )  /  (
2 ^ M ) )  =  ( N  /  ( 2  x.  ( 2 ^ M
) ) ) )
1610, 15eqtr4d 2471 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( N  /  (
2 ^ ( M  +  1 ) ) )  =  ( ( N  /  2 )  /  ( 2 ^ M ) ) )
1716fveq2d 5732 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) )  =  ( |_ `  ( ( N  / 
2 )  /  (
2 ^ M ) ) ) )
1811zred 10375 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  N  e.  RR )
1918rehalfcld 10214 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( N  /  2
)  e.  RR )
20 fldiv 11241 . . . . . 6  |-  ( ( ( N  /  2
)  e.  RR  /\  ( 2 ^ M
)  e.  NN )  ->  ( |_ `  ( ( |_ `  ( N  /  2
) )  /  (
2 ^ M ) ) )  =  ( |_ `  ( ( N  /  2 )  /  ( 2 ^ M ) ) ) )
2119, 6, 20syl2anc 643 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  (
( |_ `  ( N  /  2 ) )  /  ( 2 ^ M ) ) )  =  ( |_ `  ( ( N  / 
2 )  /  (
2 ^ M ) ) ) )
2217, 21eqtr4d 2471 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) )  =  ( |_ `  ( ( |_ `  ( N  /  2
) )  /  (
2 ^ M ) ) ) )
2322breq2d 4224 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2  ||  ( |_ `  ( N  / 
( 2 ^ ( M  +  1 ) ) ) )  <->  2  ||  ( |_ `  ( ( |_ `  ( N  /  2 ) )  /  ( 2 ^ M ) ) ) ) )
2423notbid 286 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( -.  2  ||  ( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) )  <->  -.  2  ||  ( |_
`  ( ( |_
`  ( N  / 
2 ) )  / 
( 2 ^ M
) ) ) ) )
25 peano2nn0 10260 . . 3  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
26 bitsval2 12937 . . 3  |-  ( ( N  e.  ZZ  /\  ( M  +  1
)  e.  NN0 )  ->  ( ( M  + 
1 )  e.  (bits `  N )  <->  -.  2  ||  ( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) ) ) )
2725, 26sylan2 461 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( M  + 
1 )  e.  (bits `  N )  <->  -.  2  ||  ( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) ) ) )
2819flcld 11207 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  ( N  /  2 ) )  e.  ZZ )
29 bitsval2 12937 . . 3  |-  ( ( ( |_ `  ( N  /  2 ) )  e.  ZZ  /\  M  e.  NN0 )  ->  ( M  e.  (bits `  ( |_ `  ( N  / 
2 ) ) )  <->  -.  2  ||  ( |_
`  ( ( |_
`  ( N  / 
2 ) )  / 
( 2 ^ M
) ) ) ) )
3028, 29sylancom 649 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( M  e.  (bits `  ( |_ `  ( N  /  2 ) ) )  <->  -.  2  ||  ( |_ `  ( ( |_ `  ( N  /  2 ) )  /  ( 2 ^ M ) ) ) ) )
3124, 27, 303bitr4d 277 1  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( M  + 
1 )  e.  (bits `  N )  <->  M  e.  (bits `  ( |_ `  ( N  /  2
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   1c1 8991    + caddc 8993    x. cmul 8995    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   |_cfl 11201   ^cexp 11382    || cdivides 12852  bitscbits 12931
This theorem is referenced by:  bitsp1e  12944  bitsp1o  12945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fl 11202  df-seq 11324  df-exp 11383  df-bits 12934
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