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Theorem oddpwdc 11691
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 8697 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
2  e.  CC )
3 simpr 109 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
y  e.  NN0 )
42, 3expcld 10311 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( 2 ^ y
)  e.  CC )
5 breq2 3897 . . . . . . . . . 10  |-  ( z  =  x  ->  (
2  ||  z  <->  2  ||  x ) )
65notbid 639 . . . . . . . . 9  |-  ( z  =  x  ->  ( -.  2  ||  z  <->  -.  2  ||  x ) )
7 oddpwdc.j . . . . . . . . 9  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
86, 7elrab2 2810 . . . . . . . 8  |-  ( x  e.  J  <->  ( x  e.  NN  /\  -.  2  ||  x ) )
98simplbi 270 . . . . . . 7  |-  ( x  e.  J  ->  x  e.  NN )
109adantr 272 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  NN )
1110nncnd 8638 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  CC )
124, 11mulcld 7704 . . . 4  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( ( 2 ^ y )  x.  x
)  e.  CC )
1312adantl 273 . . 3  |-  ( ( T.  /\  ( x  e.  J  /\  y  e.  NN0 ) )  -> 
( ( 2 ^ y )  x.  x
)  e.  CC )
14 nnnn0 8882 . . . . . 6  |-  ( a  e.  NN  ->  a  e.  NN0 )
15 2nn 8779 . . . . . . 7  |-  2  e.  NN
16 pw2dvdseu 11685 . . . . . . . 8  |-  ( a  e.  NN  ->  E! z  e.  NN0  ( ( 2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) )
17 riotacl 5696 . . . . . . . 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 10193 . . . . . . 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 408 . . . . . 6  |-  ( a  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) ) )  e.  NN )
21 nn0nndivcl 8937 . . . . . 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 406 . . . . 5  |-  ( a  e.  NN  ->  (
a  /  ( 2 ^ ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  a  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  a ) ) ) )  e.  RR )
2322, 18jca 302 . . . 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 273 . . 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 451 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  <->  ( (
x  e.  NN  /\  -.  2  ||  x )  /\  y  e.  NN0 ) )
2625anbi1i 451 . . . . 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 11690 . . . . 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 183 . . . 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 6084 . 2  |-  ( T. 
->  F : ( J  X.  NN0 ) -1-1-onto-> NN )
3130mptru 1321 1  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    = wceq 1312   T. wtru 1313    e. wcel 1461   E!wreu 2390   {crab 2392   class class class wbr 3893    X. cxp 4495   -1-1-onto->wf1o 5078   iota_crio 5681  (class class class)co 5726    e. cmpo 5728   CCcc 7539   RRcr 7540   1c1 7542    + caddc 7544    x. cmul 7546    / cdiv 8339   NNcn 8624   2c2 8675   NN0cn0 8875   ^cexp 10179    || cdvds 11335
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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-2 8683  df-n0 8876  df-z 8953  df-uz 9223  df-q 9308  df-rp 9338  df-fz 9678  df-fl 9930  df-mod 9983  df-seqfrec 10106  df-exp 10180  df-dvds 11336
This theorem is referenced by:  sqpweven  11692  2sqpwodd  11693  xpnnen  11746
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