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Theorem oddpwdc 12144
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 8968 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
2  e.  CC )
3 simpr 110 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
y  e.  NN0 )
42, 3expcld 10626 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( 2 ^ y
)  e.  CC )
5 breq2 4004 . . . . . . . . . 10  |-  ( z  =  x  ->  (
2  ||  z  <->  2  ||  x ) )
65notbid 667 . . . . . . . . 9  |-  ( z  =  x  ->  ( -.  2  ||  z  <->  -.  2  ||  x ) )
7 oddpwdc.j . . . . . . . . 9  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
86, 7elrab2 2896 . . . . . . . 8  |-  ( x  e.  J  <->  ( x  e.  NN  /\  -.  2  ||  x ) )
98simplbi 274 . . . . . . 7  |-  ( x  e.  J  ->  x  e.  NN )
109adantr 276 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  NN )
1110nncnd 8909 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  CC )
124, 11mulcld 7955 . . . 4  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( ( 2 ^ y )  x.  x
)  e.  CC )
1312adantl 277 . . 3  |-  ( ( T.  /\  ( x  e.  J  /\  y  e.  NN0 ) )  -> 
( ( 2 ^ y )  x.  x
)  e.  CC )
14 nnnn0 9159 . . . . . 6  |-  ( a  e.  NN  ->  a  e.  NN0 )
15 2nn 9056 . . . . . . 7  |-  2  e.  NN
16 pw2dvdseu 12138 . . . . . . . 8  |-  ( a  e.  NN  ->  E! z  e.  NN0  ( ( 2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) )
17 riotacl 5838 . . . . . . . 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 10506 . . . . . . 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 414 . . . . . 6  |-  ( a  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) ) )  e.  NN )
21 nn0nndivcl 9214 . . . . . 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 411 . . . . 5  |-  ( a  e.  NN  ->  (
a  /  ( 2 ^ ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  a  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  a ) ) ) )  e.  RR )
2322, 18jca 306 . . . 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 277 . . 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 458 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  <->  ( (
x  e.  NN  /\  -.  2  ||  x )  /\  y  e.  NN0 ) )
2625anbi1i 458 . . . . 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 12143 . . . . 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 184 . . . 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 6229 . 2  |-  ( T. 
->  F : ( J  X.  NN0 ) -1-1-onto-> NN )
3130mptru 1362 1  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    <-> wb 105    = wceq 1353   T. wtru 1354    e. wcel 2148   E!wreu 2457   {crab 2459   class class class wbr 4000    X. cxp 4620   -1-1-onto->wf1o 5210   iota_crio 5823  (class class class)co 5868    e. cmpo 5870   CCcc 7787   RRcr 7788   1c1 7790    + caddc 7792    x. cmul 7794    / cdiv 8605   NNcn 8895   2c2 8946   NN0cn0 9152   ^cexp 10492    || cdvds 11765
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-n0 9153  df-z 9230  df-uz 9505  df-q 9596  df-rp 9628  df-fz 9983  df-fl 10243  df-mod 10296  df-seqfrec 10419  df-exp 10493  df-dvds 11766
This theorem is referenced by:  sqpweven  12145  2sqpwodd  12146  xpnnen  12365
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