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Theorem oddpwdclemxy 12136
Description: Lemma for oddpwdc 12141. 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 9053 . . . . . 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 9347 . . . . . . . 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 10645 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  e.  NN )
76nnzd 9347 . . . . . . . 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 8941 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ Y )  x.  X
)  e.  NN )
108, 9eqeltrd 2252 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  A  e.  NN )
1110nnzd 9347 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  A  e.  ZZ )
126nncnd 8906 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  e.  CC )
133nncnd 8906 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  X  e.  CC )
1412, 13mulcomd 7953 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( ( 2 ^ Y )  x.  X
)  =  ( X  x.  ( 2 ^ Y ) ) )
158, 14eqtr2d 2209 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( X  x.  (
2 ^ Y ) )  =  A )
16 dvds0lem 11776 . . . . . . . 8  |-  ( ( ( X  e.  ZZ  /\  ( 2 ^ Y
)  e.  ZZ  /\  A  e.  ZZ )  /\  ( X  x.  (
2 ^ Y ) )  =  A )  ->  ( 2 ^ Y )  ||  A
)
174, 7, 11, 15, 16syl31anc 1241 . . . . . . 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 4010 . . . . . . . . . 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 9347 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
2  e.  ZZ )
216nnne0d 8937 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ Y
)  =/=  0 )
22 dvdscmulr 11795 . . . . . . . . . . 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 1242 . . . . . . . . . 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 673 . . . . . . . 8  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  -.  ( ( 2 ^ Y )  x.  2 )  ||  A )
262nncnd 8906 . . . . . . . . . 10  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
2  e.  CC )
2726, 5expp1d 10624 . . . . . . . . 9  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  -> 
( 2 ^ ( Y  +  1 ) )  =  ( ( 2 ^ Y )  x.  2 ) )
2827breq1d 4008 . . . . . . . 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 673 . . . . . . 7  |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  ->  -.  ( 2 ^ ( Y  +  1 ) )  ||  A )
30 pw2dvdseu 12135 . . . . . . . . 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 5873 . . . . . . . . . . 11  |-  ( z  =  Y  ->  (
2 ^ z )  =  ( 2 ^ Y ) )
3332breq1d 4008 . . . . . . . . . 10  |-  ( z  =  Y  ->  (
( 2 ^ z
)  ||  A  <->  ( 2 ^ Y )  ||  A ) )
34 oveq1 5872 . . . . . . . . . . . . 13  |-  ( z  =  Y  ->  (
z  +  1 )  =  ( Y  + 
1 ) )
3534oveq2d 5881 . . . . . . . . . . . 12  |-  ( z  =  Y  ->  (
2 ^ ( z  +  1 ) )  =  ( 2 ^ ( Y  +  1 ) ) )
3635breq1d 4008 . . . . . . . . . . 11  |-  ( z  =  Y  ->  (
( 2 ^ (
z  +  1 ) )  ||  A  <->  ( 2 ^ ( Y  + 
1 ) )  ||  A ) )
3736notbid 667 . . . . . . . . . 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 5843 . . . . . . . 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 943 . . . . . 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 2252 . . . . 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 10645 . . . 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 8906 . . 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 8938 . . 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 2181 . . . . . 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 5881 . . . . 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 5880 . . . 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 2209 . . 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 8772 . 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 1353    e. wcel 2146    =/= wne 2345   E!wreu 2455   class class class wbr 3998   iota_crio 5820  (class class class)co 5865   0cc0 7786   1c1 7787    + caddc 7789    x. cmul 7791    / cdiv 8602   NNcn 8892   2c2 8943   NN0cn0 9149   ZZcz 9226   ^cexp 10489    || cdvds 11762
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-n0 9150  df-z 9227  df-uz 9502  df-q 9593  df-rp 9625  df-fz 9980  df-fl 10240  df-mod 10293  df-seqfrec 10416  df-exp 10490  df-dvds 11763
This theorem is referenced by:  oddpwdclemdc  12140
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