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Theorem pw2dvdseu 11757
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 11755 . 2  |-  ( N  e.  NN  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) )
2 simpll 503 . . . . . . 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 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 ) ) )  ->  m  e.  NN0 )
4 simplrr 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 ) ) )  ->  x  e.  NN0 )
5 simprll 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 ^ m )  ||  N
)
6 simprrr 514 . . . . . . 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 11756 . . . . . 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 513 . . . . . . 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 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 ^ ( m  + 
1 ) )  ||  N )
102, 4, 3, 8, 9pw2dvdseulemle 11756 . . . . . 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 8989 . . . . . . 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 8989 . . . . . . 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 7847 . . . . . 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 913 . . . . 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 2491 . . 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 5750 . . . . . 6  |-  ( m  =  x  ->  (
2 ^ m )  =  ( 2 ^ x ) )
1817breq1d 3909 . . . . 5  |-  ( m  =  x  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ x )  ||  N ) )
19 oveq1 5749 . . . . . . . 8  |-  ( m  =  x  ->  (
m  +  1 )  =  ( x  + 
1 ) )
2019oveq2d 5758 . . . . . . 7  |-  ( m  =  x  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( x  +  1 ) ) )
2120breq1d 3909 . . . . . 6  |-  ( m  =  x  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( x  + 
1 ) )  ||  N ) )
2221notbid 641 . . . . 5  |-  ( m  =  x  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( x  +  1 ) ) 
||  N ) )
2318, 22anbi12d 464 . . . 4  |-  ( m  =  x  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )
2423rmo4 2850 . . 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 2620 . 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 413 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 1465   A.wral 2393   E.wrex 2394   E!wreu 2395   E*wrmo 2396   class class class wbr 3899  (class class class)co 5742   1c1 7589    + caddc 7591    <_ cle 7769   NNcn 8684   2c2 8735   NN0cn0 8935   ^cexp 10247    || cdvds 11405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-n0 8936  df-z 9013  df-uz 9283  df-q 9368  df-rp 9398  df-fz 9746  df-fl 9998  df-mod 10051  df-seqfrec 10174  df-exp 10248  df-dvds 11406
This theorem is referenced by:  oddpwdclemxy  11758  oddpwdclemdvds  11759  oddpwdclemndvds  11760  oddpwdclemodd  11761  oddpwdclemdc  11762  oddpwdc  11763
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