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Theorem bitsp1 12638
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 simpl 443 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  N  e.  ZZ )
21zred 10133 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  N  e.  RR )
32rehalfcld 9974 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( N  /  2
)  e.  RR )
4 2nn 9893 . . . . . . . 8  |-  2  e.  NN
54a1i 10 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
2  e.  NN )
6 simpr 447 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  M  e.  NN0 )
75, 6nnexpcld 11282 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ M
)  e.  NN )
8 fldiv 10980 . . . . . 6  |-  ( ( ( N  /  2
)  e.  RR  /\  ( 2 ^ M
)  e.  NN )  ->  ( |_ `  ( ( |_ `  ( N  /  2
) )  /  (
2 ^ M ) ) )  =  ( |_ `  ( ( N  /  2 )  /  ( 2 ^ M ) ) ) )
93, 7, 8syl2anc 642 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  (
( |_ `  ( N  /  2 ) )  /  ( 2 ^ M ) ) )  =  ( |_ `  ( ( N  / 
2 )  /  (
2 ^ M ) ) ) )
10 eqidd 2297 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  (
( |_ `  ( N  /  2 ) )  /  ( 2 ^ M ) ) )  =  ( |_ `  ( ( |_ `  ( N  /  2
) )  /  (
2 ^ M ) ) ) )
115nncnd 9778 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
2  e.  CC )
1211, 6expp1d 11262 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ ( M  +  1 ) )  =  ( ( 2 ^ M )  x.  2 ) )
137nncnd 9778 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ M
)  e.  CC )
1413, 11mulcomd 8872 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 2 ^ M )  x.  2 )  =  ( 2  x.  ( 2 ^ M ) ) )
1512, 14eqtrd 2328 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ ( M  +  1 ) )  =  ( 2  x.  ( 2 ^ M ) ) )
1615oveq2d 5890 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( N  /  (
2 ^ ( M  +  1 ) ) )  =  ( N  /  ( 2  x.  ( 2 ^ M
) ) ) )
172recnd 8877 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  N  e.  CC )
185nnne0d 9806 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
2  =/=  0 )
197nnne0d 9806 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2 ^ M
)  =/=  0 )
2017, 11, 13, 18, 19divdiv1d 9583 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( N  / 
2 )  /  (
2 ^ M ) )  =  ( N  /  ( 2  x.  ( 2 ^ M
) ) ) )
2116, 20eqtr4d 2331 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( N  /  (
2 ^ ( M  +  1 ) ) )  =  ( ( N  /  2 )  /  ( 2 ^ M ) ) )
2221fveq2d 5545 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) )  =  ( |_ `  ( ( N  / 
2 )  /  (
2 ^ M ) ) ) )
239, 10, 223eqtr4rd 2339 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) )  =  ( |_ `  ( ( |_ `  ( N  /  2
) )  /  (
2 ^ M ) ) ) )
2423breq2d 4051 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( 2  ||  ( |_ `  ( N  / 
( 2 ^ ( M  +  1 ) ) ) )  <->  2  ||  ( |_ `  ( ( |_ `  ( N  /  2 ) )  /  ( 2 ^ M ) ) ) ) )
2524notbid 285 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( -.  2  ||  ( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) )  <->  -.  2  ||  ( |_
`  ( ( |_
`  ( N  / 
2 ) )  / 
( 2 ^ M
) ) ) ) )
26 peano2nn0 10020 . . 3  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
27 bitsval2 12632 . . 3  |-  ( ( N  e.  ZZ  /\  ( M  +  1
)  e.  NN0 )  ->  ( ( M  + 
1 )  e.  (bits `  N )  <->  -.  2  ||  ( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) ) ) )
2826, 27sylan2 460 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( M  + 
1 )  e.  (bits `  N )  <->  -.  2  ||  ( |_ `  ( N  /  ( 2 ^ ( M  +  1 ) ) ) ) ) )
293flcld 10946 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( |_ `  ( N  /  2 ) )  e.  ZZ )
30 bitsval2 12632 . . 3  |-  ( ( ( |_ `  ( N  /  2 ) )  e.  ZZ  /\  M  e.  NN0 )  ->  ( M  e.  (bits `  ( |_ `  ( N  / 
2 ) ) )  <->  -.  2  ||  ( |_
`  ( ( |_
`  ( N  / 
2 ) )  / 
( 2 ^ M
) ) ) ) )
3129, 30sylancom 648 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  -> 
( M  e.  (bits `  ( |_ `  ( N  /  2 ) ) )  <->  -.  2  ||  ( |_ `  ( ( |_ `  ( N  /  2 ) )  /  ( 2 ^ M ) ) ) ) )
3225, 28, 313bitr4d 276 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   |_cfl 10940   ^cexp 11120    || cdivides 12547  bitscbits 12626
This theorem is referenced by:  bitsp1e  12639  bitsp1o  12640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fl 10941  df-seq 11063  df-exp 11121  df-bits 12629
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