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Theorem pw2dvdseu 12109
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 12107 . 2  |-  ( N  e.  NN  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) )
2 simpll 524 . . . . . . 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 530 . . . . . . 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 531 . . . . . . 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 532 . . . . . . 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 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 ) ) )  ->  -.  ( 2 ^ ( x  + 
1 ) )  ||  N )
72, 3, 4, 5, 6pw2dvdseulemle 12108 . . . . . 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 534 . . . . . . 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 533 . . . . . . 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 12108 . . . . . 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 9176 . . . . . . 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 9176 . . . . . . 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 8022 . . . . . 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 939 . . . . 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 2552 . . 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 5858 . . . . . 6  |-  ( m  =  x  ->  (
2 ^ m )  =  ( 2 ^ x ) )
1817breq1d 3997 . . . . 5  |-  ( m  =  x  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ x )  ||  N ) )
19 oveq1 5857 . . . . . . . 8  |-  ( m  =  x  ->  (
m  +  1 )  =  ( x  + 
1 ) )
2019oveq2d 5866 . . . . . . 7  |-  ( m  =  x  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( x  +  1 ) ) )
2120breq1d 3997 . . . . . 6  |-  ( m  =  x  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( x  + 
1 ) )  ||  N ) )
2221notbid 662 . . . . 5  |-  ( m  =  x  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( x  +  1 ) ) 
||  N ) )
2318, 22anbi12d 470 . . . 4  |-  ( m  =  x  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ x )  ||  N  /\  -.  ( 2 ^ ( x  + 
1 ) )  ||  N ) ) )
2423rmo4 2923 . . 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 2682 . 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 415 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 2141   A.wral 2448   E.wrex 2449   E!wreu 2450   E*wrmo 2451   class class class wbr 3987  (class class class)co 5850   1c1 7762    + caddc 7764    <_ cle 7942   NNcn 8865   2c2 8916   NN0cn0 9122   ^cexp 10462    || cdvds 11736
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fl 10213  df-mod 10266  df-seqfrec 10389  df-exp 10463  df-dvds 11737
This theorem is referenced by:  oddpwdclemxy  12110  oddpwdclemdvds  12111  oddpwdclemndvds  12112  oddpwdclemodd  12113  oddpwdclemdc  12114  oddpwdc  12115
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