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Theorem oddpwdc 11234
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 8466 . . . . . 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 10051 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( 2 ^ y
)  e.  CC )
5 breq2 3841 . . . . . . . . . 10  |-  ( z  =  x  ->  (
2  ||  z  <->  2  ||  x ) )
65notbid 627 . . . . . . . . 9  |-  ( z  =  x  ->  ( -.  2  ||  z  <->  -.  2  ||  x ) )
7 oddpwdc.j . . . . . . . . 9  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
86, 7elrab2 2772 . . . . . . . 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 8408 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  CC )
124, 11mulcld 7487 . . . 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 8650 . . . . . 6  |-  ( a  e.  NN  ->  a  e.  NN0 )
15 2nn 8547 . . . . . . 7  |-  2  e.  NN
16 pw2dvdseu 11228 . . . . . . . 8  |-  ( a  e.  NN  ->  E! z  e.  NN0  ( ( 2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) )
17 riotacl 5604 . . . . . . . 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 9933 . . . . . . 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 8705 . . . . . 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 11233 . . . . 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 5982 . 2  |-  ( T. 
->  F : ( J  X.  NN0 ) -1-1-onto-> NN )
3130mptru 1298 1  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    = wceq 1289   T. wtru 1290    e. wcel 1438   E!wreu 2361   {crab 2363   class class class wbr 3837    X. cxp 4426   -1-1-onto->wf1o 5001   iota_crio 5589  (class class class)co 5634    |-> cmpt2 5636   CCcc 7327   RRcr 7328   1c1 7330    + caddc 7332    x. cmul 7334    / cdiv 8113   NNcn 8394   2c2 8444   NN0cn0 8643   ^cexp 9919    || cdvds 10878
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-dvds 10879
This theorem is referenced by:  sqpweven  11235  2sqpwodd  11236  xpnnen  11289
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