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Theorem oddpwdc 10696
Description: The function  F that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.)
Hypotheses
Ref Expression
oddpwdc.j  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
oddpwdc.f  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
Assertion
Ref Expression
oddpwdc  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
Distinct variable groups:    x, y, z   
x, J, y
Allowed substitution hints:    F( x, y, z)    J( z)

Proof of Theorem oddpwdc
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 oddpwdc.f . . 3  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
2 2cnd 8179 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
2  e.  CC )
3 simpr 108 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
y  e.  NN0 )
42, 3expcld 9702 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( 2 ^ y
)  e.  CC )
5 breq2 3797 . . . . . . . . . 10  |-  ( z  =  x  ->  (
2  ||  z  <->  2  ||  x ) )
65notbid 625 . . . . . . . . 9  |-  ( z  =  x  ->  ( -.  2  ||  z  <->  -.  2  ||  x ) )
7 oddpwdc.j . . . . . . . . 9  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
86, 7elrab2 2752 . . . . . . . 8  |-  ( x  e.  J  <->  ( x  e.  NN  /\  -.  2  ||  x ) )
98simplbi 268 . . . . . . 7  |-  ( x  e.  J  ->  x  e.  NN )
109adantr 270 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  NN )
1110nncnd 8120 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  CC )
124, 11mulcld 7201 . . . 4  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( ( 2 ^ y )  x.  x
)  e.  CC )
1312adantl 271 . . 3  |-  ( ( T.  /\  ( x  e.  J  /\  y  e.  NN0 ) )  -> 
( ( 2 ^ y )  x.  x
)  e.  CC )
14 nnnn0 8362 . . . . . 6  |-  ( a  e.  NN  ->  a  e.  NN0 )
15 2nn 8260 . . . . . . 7  |-  2  e.  NN
16 pw2dvdseu 10690 . . . . . . . 8  |-  ( a  e.  NN  ->  E! z  e.  NN0  ( ( 2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) )
17 riotacl 5513 . . . . . . . 8  |-  ( E! z  e.  NN0  (
( 2 ^ z
)  ||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a )  ->  ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  a  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  a ) )  e. 
NN0 )
1816, 17syl 14 . . . . . . 7  |-  ( a  e.  NN  ->  ( iota_ z  e.  NN0  (
( 2 ^ z
)  ||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) )  e.  NN0 )
19 nnexpcl 9586 . . . . . . 7  |-  ( ( 2  e.  NN  /\  ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  a  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  a
) )  e.  NN0 )  ->  ( 2 ^ ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  a  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  a
) ) )  e.  NN )
2015, 18, 19sylancr 405 . . . . . 6  |-  ( a  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) ) )  e.  NN )
21 nn0nndivcl 8417 . . . . . 6  |-  ( ( a  e.  NN0  /\  ( 2 ^ ( iota_ z  e.  NN0  (
( 2 ^ z
)  ||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) ) )  e.  NN )  ->  ( a  / 
( 2 ^ ( iota_ z  e.  NN0  (
( 2 ^ z
)  ||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) ) ) )  e.  RR )
2214, 20, 21syl2anc 403 . . . . 5  |-  ( a  e.  NN  ->  (
a  /  ( 2 ^ ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  a  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  a ) ) ) )  e.  RR )
2322, 18jca 300 . . . 4  |-  ( a  e.  NN  ->  (
( a  /  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) ) ) )  e.  RR  /\  ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) )  e.  NN0 )
)
2423adantl 271 . . 3  |-  ( ( T.  /\  a  e.  NN )  ->  (
( a  /  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) ) ) )  e.  RR  /\  ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) )  e.  NN0 )
)
258anbi1i 446 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  <->  ( (
x  e.  NN  /\  -.  2  ||  x )  /\  y  e.  NN0 ) )
2625anbi1i 446 . . . . 5  |-  ( ( ( x  e.  J  /\  y  e.  NN0 )  /\  a  =  ( ( 2 ^ y
)  x.  x ) )  <->  ( ( ( x  e.  NN  /\  -.  2  ||  x )  /\  y  e.  NN0 )  /\  a  =  ( ( 2 ^ y
)  x.  x ) ) )
27 oddpwdclemdc 10695 . . . . 5  |-  ( ( ( ( x  e.  NN  /\  -.  2  ||  x )  /\  y  e.  NN0 )  /\  a  =  ( ( 2 ^ y )  x.  x ) )  <->  ( a  e.  NN  /\  ( 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 ) ) ) ) )
2826, 27bitri 182 . . . 4  |-  ( ( ( x  e.  J  /\  y  e.  NN0 )  /\  a  =  ( ( 2 ^ y
)  x.  x ) )  <->  ( a  e.  NN  /\  ( 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 ) ) ) ) )
2928a1i 9 . . 3  |-  ( T. 
->  ( ( ( x  e.  J  /\  y  e.  NN0 )  /\  a  =  ( ( 2 ^ y )  x.  x ) )  <->  ( a  e.  NN  /\  ( 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 ) ) ) ) ) )
301, 13, 24, 29f1od2 5887 . 2  |-  ( T. 
->  F : ( J  X.  NN0 ) -1-1-onto-> NN )
3130trud 1294 1  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    = wceq 1285   T. wtru 1286    e. wcel 1434   E!wreu 2351   {crab 2353   class class class wbr 3793    X. cxp 4369   -1-1-onto->wf1o 4931   iota_crio 5498  (class class class)co 5543    |-> cmpt2 5545   CCcc 7041   RRcr 7042   1c1 7044    + caddc 7046    x. cmul 7048    / cdiv 7827   NNcn 8106   2c2 8156   NN0cn0 8355   ^cexp 9572    || cdvds 10340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-n0 8356  df-z 8433  df-uz 8701  df-q 8786  df-rp 8816  df-fz 9106  df-fl 9352  df-mod 9405  df-iseq 9522  df-iexp 9573  df-dvds 10341
This theorem is referenced by:  sqpweven  10697  2sqpwodd  10698  xpnnen  10705
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