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Theorem pw2dvdseu 11685
Description: A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.)
Assertion
Ref Expression
pw2dvdseu  |-  ( N  e.  NN  ->  E! m  e.  NN0  ( ( 2 ^ m ) 
||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N ) )
Distinct variable group:    m, N

Proof of Theorem pw2dvdseu
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pw2dvds 11683 . 2  |-  ( N  e.  NN  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) )
2 simpll 501 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  N  e.  NN )
3 simplrl 507 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  m  e.  NN0 )
4 simplrr 508 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  x  e.  NN0 )
5 simprll 509 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  ( 2 ^ m )  ||  N
)
6 simprrr 512 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  -.  ( 2 ^ ( x  + 
1 ) )  ||  N )
72, 3, 4, 5, 6pw2dvdseulemle 11684 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  m  <_  x
)
8 simprrl 511 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  ( 2 ^ x )  ||  N
)
9 simprlr 510 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  -.  ( 2 ^ ( m  + 
1 ) )  ||  N )
102, 4, 3, 8, 9pw2dvdseulemle 11684 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  x  <_  m
)
113nn0red 8929 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  m  e.  RR )
124nn0red 8929 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  x  e.  RR )
1311, 12letri3d 7796 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  ( m  =  x  <->  ( m  <_  x  /\  x  <_  m
) ) )
147, 10, 13mpbir2and 909 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  /\  ( (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )  ->  m  =  x )
1514ex 114 . . . 4  |-  ( ( N  e.  NN  /\  ( m  e.  NN0  /\  x  e.  NN0 )
)  ->  ( (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  /\  ( (
2 ^ x ) 
||  N  /\  -.  ( 2 ^ (
x  +  1 ) )  ||  N ) )  ->  m  =  x ) )
1615ralrimivva 2486 . . 3  |-  ( N  e.  NN  ->  A. m  e.  NN0  A. x  e. 
NN0  ( ( ( ( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  /\  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) )  ->  m  =  x )
)
17 oveq2 5734 . . . . . 6  |-  ( m  =  x  ->  (
2 ^ m )  =  ( 2 ^ x ) )
1817breq1d 3903 . . . . 5  |-  ( m  =  x  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ x )  ||  N ) )
19 oveq1 5733 . . . . . . . 8  |-  ( m  =  x  ->  (
m  +  1 )  =  ( x  + 
1 ) )
2019oveq2d 5742 . . . . . . 7  |-  ( m  =  x  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( x  +  1 ) ) )
2120breq1d 3903 . . . . . 6  |-  ( m  =  x  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( x  + 
1 ) )  ||  N ) )
2221notbid 639 . . . . 5  |-  ( m  =  x  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( x  +  1 ) ) 
||  N ) )
2318, 22anbi12d 462 . . . 4  |-  ( m  =  x  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )
2423rmo4 2844 . . 3  |-  ( E* m  e.  NN0  (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  <->  A. m  e.  NN0  A. x  e.  NN0  (
( ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N )  /\  (
( 2 ^ x
)  ||  N  /\  -.  ( 2 ^ (
x  +  1 ) )  ||  N ) )  ->  m  =  x ) )
2516, 24sylibr 133 . 2  |-  ( N  e.  NN  ->  E* m  e.  NN0  ( ( 2 ^ m ) 
||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N ) )
26 reu5 2615 . 2  |-  ( E! m  e.  NN0  (
( 2 ^ m
)  ||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N )  <-> 
( E. m  e. 
NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N )  /\  E* m  e.  NN0  ( ( 2 ^ m ) 
||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N ) ) )
271, 25, 26sylanbrc 411 1  |-  ( N  e.  NN  ->  E! m  e.  NN0  ( ( 2 ^ m ) 
||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 1461   A.wral 2388   E.wrex 2389   E!wreu 2390   E*wrmo 2391   class class class wbr 3893  (class class class)co 5726   1c1 7542    + caddc 7544    <_ cle 7719   NNcn 8624   2c2 8675   NN0cn0 8875   ^cexp 10179    || cdvds 11335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-2 8683  df-n0 8876  df-z 8953  df-uz 9223  df-q 9308  df-rp 9338  df-fz 9678  df-fl 9930  df-mod 9983  df-seqfrec 10106  df-exp 10180  df-dvds 11336
This theorem is referenced by:  oddpwdclemxy  11686  oddpwdclemdvds  11687  oddpwdclemndvds  11688  oddpwdclemodd  11689  oddpwdclemdc  11690  oddpwdc  11691
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