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Theorem oddpwdclemxy 11639
Description: Lemma for oddpwdc 11644. Another way of stating that decomposing a natural number into a power of two and an odd number is unique. (Contributed by Jim Kingdon, 16-Nov-2021.)
Assertion
Ref Expression
oddpwdclemxy  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( X  =  ( A  /  ( 2 ^ ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) ) ) )  /\  Y  =  ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) ) )
Distinct variable groups:    z, A    z, Y
Allowed substitution hint:    X( z)

Proof of Theorem oddpwdclemxy
StepHypRef Expression
1 2nn 8733 . . . . . 6  |-  2  e.  NN
21a1i 9 . . . . 5  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
2  e.  NN )
3 simplll 503 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  X  e.  NN )
43nnzd 9024 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  X  e.  ZZ )
5 simplr 500 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  Y  e.  NN0 )
62, 5nnexpcld 10287 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  e.  NN )
76nnzd 9024 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  e.  ZZ )
8 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  A  =  ( (
2 ^ Y )  x.  X ) )
96, 3nnmulcld 8627 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ Y )  x.  X
)  e.  NN )
108, 9eqeltrd 2176 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  A  e.  NN )
1110nnzd 9024 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  A  e.  ZZ )
126nncnd 8592 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  e.  CC )
133nncnd 8592 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  X  e.  CC )
1412, 13mulcomd 7659 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ Y )  x.  X
)  =  ( X  x.  ( 2 ^ Y ) ) )
158, 14eqtr2d 2133 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( X  x.  (
2 ^ Y ) )  =  A )
16 dvds0lem 11298 . . . . . . . 8  |-  ( ( ( X  e.  ZZ  /\  ( 2 ^ Y
)  e.  ZZ  /\  A  e.  ZZ )  /\  ( X  x.  (
2 ^ Y ) )  =  A )  ->  ( 2 ^ Y )  ||  A
)
174, 7, 11, 15, 16syl31anc 1187 . . . . . . 7  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  ||  A )
18 simpllr 504 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  -.  2  ||  X )
198breq2d 3887 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( ( 2 ^ Y )  x.  2 )  ||  A  <->  ( ( 2 ^ Y
)  x.  2 ) 
||  ( ( 2 ^ Y )  x.  X ) ) )
202nnzd 9024 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
2  e.  ZZ )
216nnne0d 8623 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  =/=  0 )
22 dvdscmulr 11317 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  X  e.  ZZ  /\  (
( 2 ^ Y
)  e.  ZZ  /\  ( 2 ^ Y
)  =/=  0 ) )  ->  ( (
( 2 ^ Y
)  x.  2 ) 
||  ( ( 2 ^ Y )  x.  X )  <->  2  ||  X ) )
2320, 4, 7, 21, 22syl112anc 1188 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( ( 2 ^ Y )  x.  2 )  ||  (
( 2 ^ Y
)  x.  X )  <->  2  ||  X ) )
2419, 23bitrd 187 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( ( 2 ^ Y )  x.  2 )  ||  A  <->  2 
||  X ) )
2518, 24mtbird 639 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  -.  ( ( 2 ^ Y )  x.  2 )  ||  A )
262nncnd 8592 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
2  e.  CC )
2726, 5expp1d 10266 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ ( Y  +  1 ) )  =  ( ( 2 ^ Y )  x.  2 ) )
2827breq1d 3885 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ ( Y  +  1 ) )  ||  A  <->  ( ( 2 ^ Y
)  x.  2 ) 
||  A ) )
2925, 28mtbird 639 . . . . . . 7  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  -.  ( 2 ^ ( Y  +  1 ) )  ||  A )
30 pw2dvdseu 11638 . . . . . . . . 9  |-  ( A  e.  NN  ->  E! z  e.  NN0  ( ( 2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) )
3110, 30syl 14 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  E! z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) )
32 oveq2 5714 . . . . . . . . . . 11  |-  ( z  =  Y  ->  (
2 ^ z )  =  ( 2 ^ Y ) )
3332breq1d 3885 . . . . . . . . . 10  |-  ( z  =  Y  ->  (
( 2 ^ z
)  ||  A  <->  ( 2 ^ Y )  ||  A ) )
34 oveq1 5713 . . . . . . . . . . . . 13  |-  ( z  =  Y  ->  (
z  +  1 )  =  ( Y  + 
1 ) )
3534oveq2d 5722 . . . . . . . . . . . 12  |-  ( z  =  Y  ->  (
2 ^ ( z  +  1 ) )  =  ( 2 ^ ( Y  +  1 ) ) )
3635breq1d 3885 . . . . . . . . . . 11  |-  ( z  =  Y  ->  (
( 2 ^ (
z  +  1 ) )  ||  A  <->  ( 2 ^ ( Y  + 
1 ) )  ||  A ) )
3736notbid 633 . . . . . . . . . 10  |-  ( z  =  Y  ->  ( -.  ( 2 ^ (
z  +  1 ) )  ||  A  <->  -.  (
2 ^ ( Y  +  1 ) ) 
||  A ) )
3833, 37anbi12d 460 . . . . . . . . 9  |-  ( z  =  Y  ->  (
( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
)  <->  ( ( 2 ^ Y )  ||  A  /\  -.  ( 2 ^ ( Y  + 
1 ) )  ||  A ) ) )
3938riota2 5684 . . . . . . . 8  |-  ( ( Y  e.  NN0  /\  E! z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) )  ->  (
( ( 2 ^ Y )  ||  A  /\  -.  ( 2 ^ ( Y  +  1 ) )  ||  A
)  <->  ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) )  =  Y ) )
405, 31, 39syl2anc 406 . . . . . . 7  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( ( 2 ^ Y )  ||  A  /\  -.  ( 2 ^ ( Y  + 
1 ) )  ||  A )  <->  ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) )  =  Y ) )
4117, 29, 40mpbi2and 895 . . . . . 6  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) )  =  Y )
4241, 5eqeltrd 2176 . . . . 5  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) )  e.  NN0 )
432, 42nnexpcld 10287 . . . 4  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ ( iota_ z  e.  NN0  (
( 2 ^ z
)  ||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  e.  NN )
4443nncnd 8592 . . 3  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ ( iota_ z  e.  NN0  (
( 2 ^ z
)  ||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  e.  CC )
4543nnap0d 8624 . . 3  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ ( iota_ z  e.  NN0  (
( 2 ^ z
)  ||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) ) #  0 )
4641eqcomd 2105 . . . . . 6  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  Y  =  ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )
4746oveq2d 5722 . . . . 5  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  =  ( 2 ^ ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) ) ) )
4847oveq1d 5721 . . . 4  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ Y )  x.  X
)  =  ( ( 2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  x.  X
) )
498, 48eqtr2d 2133 . . 3  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) )  x.  X )  =  A )
5044, 13, 45, 49mvllmulapd 8459 . 2  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  X  =  ( A  /  ( 2 ^ ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) ) ) )
5150, 46jca 302 1  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( X  =  ( A  /  ( 2 ^ ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) ) ) )  /\  Y  =  ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448    =/= wne 2267   E!wreu 2377   class class class wbr 3875   iota_crio 5661  (class class class)co 5706   0cc0 7500   1c1 7501    + caddc 7503    x. cmul 7505    / cdiv 8293   NNcn 8578   2c2 8629   NN0cn0 8829   ZZcz 8906   ^cexp 10133    || cdvds 11288
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-dvds 11289
This theorem is referenced by:  oddpwdclemdc  11643
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