Theorem oddpwdc 12691

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.

Step	Hyp	Ref	Expression
1		oddpwdc.f	. . 3 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
2		2cnd 9179	. . . . . 6 |- ( ( x e. J /\ y e. NN0 ) -> 2 e. CC )
3		simpr 110	. . . . . 6 |- ( ( x e. J /\ y e. NN0 ) -> y e. NN0 )
4	2, 3	expcld 10890	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> ( 2 ^ y ) e. CC )
5		breq2 4086	. . . . . . . . . 10 |- ( z = x -> ( 2 || z <-> 2 || x ) )
6	5	notbid 671	. . . . . . . . 9 |- ( z = x -> ( -. 2 || z <-> -. 2 || x ) )
7		oddpwdc.j	. . . . . . . . 9 |- J = { z e. NN | -. 2 || z }
8	6, 7	elrab2 2962	. . . . . . . 8 |- ( x e. J <-> ( x e. NN /\ -. 2 || x ) )
9	8	simplbi 274	. . . . . . 7 |- ( x e. J -> x e. NN )
10	9	adantr 276	. . . . . 6 |- ( ( x e. J /\ y e. NN0 ) -> x e. NN )
11	10	nncnd 9120	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> x e. CC )
12	4, 11	mulcld 8163	. . . 4 |- ( ( x e. J /\ y e. NN0 ) -> ( ( 2 ^ y ) x. x ) e. CC )
13	12	adantl 277	. . 3 |- ( ( T. /\ ( x e. J /\ y e. NN0 ) ) -> ( ( 2 ^ y ) x. x ) e. CC )
14		nnnn0 9372	. . . . . 6 |- ( a e. NN -> a e. NN0 )
15		2nn 9268	. . . . . . 7 |- 2 e. NN
16		pw2dvdseu 12685	. . . . . . . 8 |- ( a e. NN -> E! z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) )
17		riotacl 5969	. . . . . . . 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 )
18	16, 17	syl 14	. . . . . . 7 |- ( a e. NN -> ( iota_ z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) ) e. NN0 )
19		nnexpcl 10769	. . . . . . 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 )
20	15, 18, 19	sylancr 414	. . . . . 6 |- ( a e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) ) ) e. NN )
21		nn0nndivcl 9427	. . . . . 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 )
22	14, 20, 21	syl2anc 411	. . . . 5 |- ( a e. NN -> ( a / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) ) ) ) e. RR )
23	22, 18	jca 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 ) )
24	23	adantl 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 ) )
25	8	anbi1i 458	. . . . . 6 |- ( ( x e. J /\ y e. NN0 ) <-> ( ( x e. NN /\ -. 2 || x ) /\ y e. NN0 ) )
26	25	anbi1i 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 12690	. . . . 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 ) ) ) ) )
28	26, 27	bitri 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 ) ) ) ) )
29	28	a1i 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 ) ) ) ) ) )
30	1, 13, 24, 29	f1od2 6379	. 2 |- ( T. -> F : ( J X. NN0 ) -1-1-onto-> NN )
31	30	mptru 1404	1 |- F : ( J X. NN0 ) -1-1-onto-> NN

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1395   T. wtru 1396   e. wcel 2200   E!wreu 2510   {crab 2512   class class class wbr 4082   X. cxp 4716   -1-1-onto->wf1o 5316   iota_crio 5952  (class class class)co 6000   e. cmpo 6002   CCcc 7993   RRcr 7994   1c1 7996   + caddc 7998   x. cmul 8000   / cdiv 8815   NNcn 9106   2c2 9157   NN0cn0 9365   ^cexp 10755   || cdvds 12293

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-dvds 12294

This theorem is referenced by:  sqpweven  12692  2sqpwodd  12693  xpnnen  12960