Theorem oddpwdc 12312

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 9055	. . . . . 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 10744	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> ( 2 ^ y ) e. CC )
5		breq2 4033	. . . . . . . . . 10 |- ( z = x -> ( 2 || z <-> 2 || x ) )
6	5	notbid 668	. . . . . . . . 9 |- ( z = x -> ( -. 2 || z <-> -. 2 || x ) )
7		oddpwdc.j	. . . . . . . . 9 |- J = { z e. NN | -. 2 || z }
8	6, 7	elrab2 2919	. . . . . . . 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 8996	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> x e. CC )
12	4, 11	mulcld 8040	. . . 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 9247	. . . . . 6 |- ( a e. NN -> a e. NN0 )
15		2nn 9143	. . . . . . 7 |- 2 e. NN
16		pw2dvdseu 12306	. . . . . . . 8 |- ( a e. NN -> E! z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) )
17		riotacl 5888	. . . . . . . 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 10623	. . . . . . 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 9302	. . . . . 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 12311	. . . . 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 6288	. 2 |- ( T. -> F : ( J X. NN0 ) -1-1-onto-> NN )
31	30	mptru 1373	1 |- F : ( J X. NN0 ) -1-1-onto-> NN

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1364   T. wtru 1365   e. wcel 2164   E!wreu 2474   {crab 2476   class class class wbr 4029   X. cxp 4657   -1-1-onto->wf1o 5253   iota_crio 5872  (class class class)co 5918   e. cmpo 5920   CCcc 7870   RRcr 7871   1c1 7873   + caddc 7875   x. cmul 7877   / cdiv 8691   NNcn 8982   2c2 9033   NN0cn0 9240   ^cexp 10609   || cdvds 11930

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-dvds 11931

This theorem is referenced by:  sqpweven  12313  2sqpwodd  12314  xpnnen  12551