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Theorem pw2dvdseu 12199
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 12197 . 2  |-  ( N  e.  NN  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) )
2 simpll 527 . . . . . . 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 535 . . . . . . 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 536 . . . . . . 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 537 . . . . . . 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 540 . . . . . . 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 12198 . . . . . 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 539 . . . . . . 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 538 . . . . . . 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 12198 . . . . . 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 9259 . . . . . . 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 9259 . . . . . . 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 8102 . . . . . 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 946 . . . . 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 115 . . . 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 2572 . . 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 5903 . . . . . 6  |-  ( m  =  x  ->  (
2 ^ m )  =  ( 2 ^ x ) )
1817breq1d 4028 . . . . 5  |-  ( m  =  x  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ x )  ||  N ) )
19 oveq1 5902 . . . . . . . 8  |-  ( m  =  x  ->  (
m  +  1 )  =  ( x  + 
1 ) )
2019oveq2d 5911 . . . . . . 7  |-  ( m  =  x  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( x  +  1 ) ) )
2120breq1d 4028 . . . . . 6  |-  ( m  =  x  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( x  + 
1 ) )  ||  N ) )
2221notbid 668 . . . . 5  |-  ( m  =  x  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( x  +  1 ) ) 
||  N ) )
2318, 22anbi12d 473 . . . 4  |-  ( m  =  x  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )
2423rmo4 2945 . . 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 134 . 2  |-  ( N  e.  NN  ->  E* m  e.  NN0  ( ( 2 ^ m ) 
||  N  /\  -.  ( 2 ^ (
m  +  1 ) )  ||  N ) )
26 reu5 2703 . 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 417 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 104    e. wcel 2160   A.wral 2468   E.wrex 2469   E!wreu 2470   E*wrmo 2471   class class class wbr 4018  (class class class)co 5895   1c1 7841    + caddc 7843    <_ cle 8022   NNcn 8948   2c2 8999   NN0cn0 9205   ^cexp 10549    || cdvds 11825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-fz 10038  df-fl 10300  df-mod 10353  df-seqfrec 10476  df-exp 10550  df-dvds 11826
This theorem is referenced by:  oddpwdclemxy  12200  oddpwdclemdvds  12201  oddpwdclemndvds  12202  oddpwdclemodd  12203  oddpwdclemdc  12204  oddpwdc  12205
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