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Theorem oddpwdclemxy 12200
Description: Lemma for oddpwdc 12205. 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 9109 . . . . . 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 533 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  X  e.  NN )
43nnzd 9403 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  X  e.  ZZ )
5 simplr 528 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  Y  e.  NN0 )
62, 5nnexpcld 10706 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  e.  NN )
76nnzd 9403 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  e.  ZZ )
8 simpr 110 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  A  =  ( (
2 ^ Y )  x.  X ) )
96, 3nnmulcld 8997 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ Y )  x.  X
)  e.  NN )
108, 9eqeltrd 2266 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  A  e.  NN )
1110nnzd 9403 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  A  e.  ZZ )
126nncnd 8962 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  e.  CC )
133nncnd 8962 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  X  e.  CC )
1412, 13mulcomd 8008 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ Y )  x.  X
)  =  ( X  x.  ( 2 ^ Y ) ) )
158, 14eqtr2d 2223 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( X  x.  (
2 ^ Y ) )  =  A )
16 dvds0lem 11839 . . . . . . . 8  |-  ( ( ( X  e.  ZZ  /\  ( 2 ^ Y
)  e.  ZZ  /\  A  e.  ZZ )  /\  ( X  x.  (
2 ^ Y ) )  =  A )  ->  ( 2 ^ Y )  ||  A
)
174, 7, 11, 15, 16syl31anc 1252 . . . . . . 7  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  ||  A )
18 simpllr 534 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  -.  2  ||  X )
198breq2d 4030 . . . . . . . . . 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 9403 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
2  e.  ZZ )
216nnne0d 8993 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  =/=  0 )
22 dvdscmulr 11858 . . . . . . . . . . 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 1253 . . . . . . . . . 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 188 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( ( 2 ^ Y )  x.  2 )  ||  A  <->  2 
||  X ) )
2518, 24mtbird 674 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  -.  ( ( 2 ^ Y )  x.  2 )  ||  A )
262nncnd 8962 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
2  e.  CC )
2726, 5expp1d 10685 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ ( Y  +  1 ) )  =  ( ( 2 ^ Y )  x.  2 ) )
2827breq1d 4028 . . . . . . . 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 674 . . . . . . 7  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  -.  ( 2 ^ ( Y  +  1 ) )  ||  A )
30 pw2dvdseu 12199 . . . . . . . . 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 5903 . . . . . . . . . . 11  |-  ( z  =  Y  ->  (
2 ^ z )  =  ( 2 ^ Y ) )
3332breq1d 4028 . . . . . . . . . 10  |-  ( z  =  Y  ->  (
( 2 ^ z
)  ||  A  <->  ( 2 ^ Y )  ||  A ) )
34 oveq1 5902 . . . . . . . . . . . . 13  |-  ( z  =  Y  ->  (
z  +  1 )  =  ( Y  + 
1 ) )
3534oveq2d 5911 . . . . . . . . . . . 12  |-  ( z  =  Y  ->  (
2 ^ ( z  +  1 ) )  =  ( 2 ^ ( Y  +  1 ) ) )
3635breq1d 4028 . . . . . . . . . . 11  |-  ( z  =  Y  ->  (
( 2 ^ (
z  +  1 ) )  ||  A  <->  ( 2 ^ ( Y  + 
1 ) )  ||  A ) )
3736notbid 668 . . . . . . . . . 10  |-  ( z  =  Y  ->  ( -.  ( 2 ^ (
z  +  1 ) )  ||  A  <->  -.  (
2 ^ ( Y  +  1 ) ) 
||  A ) )
3833, 37anbi12d 473 . . . . . . . . 9  |-  ( z  =  Y  ->  (
( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
)  <->  ( ( 2 ^ Y )  ||  A  /\  -.  ( 2 ^ ( Y  + 
1 ) )  ||  A ) ) )
3938riota2 5873 . . . . . . . 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 411 . . . . . . 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 945 . . . . . 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 2266 . . . . 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 10706 . . . 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 8962 . . 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 8994 . . 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 2195 . . . . . 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 5911 . . . . 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 5910 . . . 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 2223 . . 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 8828 . 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 306 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 104    <-> wb 105    = wceq 1364    e. wcel 2160    =/= wne 2360   E!wreu 2470   class class class wbr 4018   iota_crio 5850  (class class class)co 5895   0cc0 7840   1c1 7841    + caddc 7843    x. cmul 7845    / cdiv 8658   NNcn 8948   2c2 8999   NN0cn0 9205   ZZcz 9282   ^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:  oddpwdclemdc  12204
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