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Theorem oddpwdc 11779
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 8761 . . . . . 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 10392 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( 2 ^ y
)  e.  CC )
5 breq2 3903 . . . . . . . . . 10  |-  ( z  =  x  ->  (
2  ||  z  <->  2  ||  x ) )
65notbid 641 . . . . . . . . 9  |-  ( z  =  x  ->  ( -.  2  ||  z  <->  -.  2  ||  x ) )
7 oddpwdc.j . . . . . . . . 9  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
86, 7elrab2 2816 . . . . . . . 8  |-  ( x  e.  J  <->  ( x  e.  NN  /\  -.  2  ||  x ) )
98simplbi 272 . . . . . . 7  |-  ( x  e.  J  ->  x  e.  NN )
109adantr 274 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  NN )
1110nncnd 8702 . . . . 5  |-  ( ( x  e.  J  /\  y  e.  NN0 )  ->  x  e.  CC )
124, 11mulcld 7754 . . . 4  |-  ( ( x  e.  J  /\  y  e.  NN0 )  -> 
( ( 2 ^ y )  x.  x
)  e.  CC )
1312adantl 275 . . 3  |-  ( ( T.  /\  ( x  e.  J  /\  y  e.  NN0 ) )  -> 
( ( 2 ^ y )  x.  x
)  e.  CC )
14 nnnn0 8952 . . . . . 6  |-  ( a  e.  NN  ->  a  e.  NN0 )
15 2nn 8849 . . . . . . 7  |-  2  e.  NN
16 pw2dvdseu 11773 . . . . . . . 8  |-  ( a  e.  NN  ->  E! z  e.  NN0  ( ( 2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) )
17 riotacl 5712 . . . . . . . 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 10274 . . . . . . 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 410 . . . . . 6  |-  ( a  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  a  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  a ) ) )  e.  NN )
21 nn0nndivcl 9007 . . . . . 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 408 . . . . 5  |-  ( a  e.  NN  ->  (
a  /  ( 2 ^ ( iota_ z  e. 
NN0  ( ( 2 ^ z )  ||  a  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  a ) ) ) )  e.  RR )
2322, 18jca 304 . . . 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 275 . . 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 453 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  NN0 )  <->  ( (
x  e.  NN  /\  -.  2  ||  x )  /\  y  e.  NN0 ) )
2625anbi1i 453 . . . . 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 11778 . . . . 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 6100 . 2  |-  ( T. 
->  F : ( J  X.  NN0 ) -1-1-onto-> NN )
3130mptru 1325 1  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    = wceq 1316   T. wtru 1317    e. wcel 1465   E!wreu 2395   {crab 2397   class class class wbr 3899    X. cxp 4507   -1-1-onto->wf1o 5092   iota_crio 5697  (class class class)co 5742    e. cmpo 5744   CCcc 7586   RRcr 7587   1c1 7589    + caddc 7591    x. cmul 7593    / cdiv 8400   NNcn 8688   2c2 8739   NN0cn0 8945   ^cexp 10260    || cdvds 11420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-dvds 11421
This theorem is referenced by:  sqpweven  11780  2sqpwodd  11781  xpnnen  11834
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