Theorem oddpwdc 12611

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 9144	. . . . . 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 10855	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> ( 2 ^ y ) e. CC )
5		breq2 4063	. . . . . . . . . 10 |- ( z = x -> ( 2 || z <-> 2 || x ) )
6	5	notbid 669	. . . . . . . . 9 |- ( z = x -> ( -. 2 || z <-> -. 2 || x ) )
7		oddpwdc.j	. . . . . . . . 9 |- J = { z e. NN | -. 2 || z }
8	6, 7	elrab2 2939	. . . . . . . 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 9085	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> x e. CC )
12	4, 11	mulcld 8128	. . . 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 9337	. . . . . 6 |- ( a e. NN -> a e. NN0 )
15		2nn 9233	. . . . . . 7 |- 2 e. NN
16		pw2dvdseu 12605	. . . . . . . 8 |- ( a e. NN -> E! z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) )
17		riotacl 5937	. . . . . . . 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 10734	. . . . . . 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 9392	. . . . . 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 12610	. . . . 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 6344	. 2 |- ( T. -> F : ( J X. NN0 ) -1-1-onto-> NN )
31	30	mptru 1382	1 |- F : ( J X. NN0 ) -1-1-onto-> NN

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1373   T. wtru 1374   e. wcel 2178   E!wreu 2488   {crab 2490   class class class wbr 4059   X. cxp 4691   -1-1-onto->wf1o 5289   iota_crio 5921  (class class class)co 5967   e. cmpo 5969   CCcc 7958   RRcr 7959   1c1 7961   + caddc 7963   x. cmul 7965   / cdiv 8780   NNcn 9071   2c2 9122   NN0cn0 9330   ^cexp 10720   || cdvds 12213

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-dvds 12214

This theorem is referenced by:  sqpweven  12612  2sqpwodd  12613  xpnnen  12880