Theorem oddpwdc 12952

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 9377	. . . . . 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 11111	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> ( 2 ^ y ) e. CC )
5		breq2 4134	. . . . . . . . . 10 |- ( z = x -> ( 2 || z <-> 2 || x ) )
6	5	notbid 677	. . . . . . . . 9 |- ( z = x -> ( -. 2 || z <-> -. 2 || x ) )
7		oddpwdc.j	. . . . . . . . 9 |- J = { z e. NN | -. 2 || z }
8	6, 7	elrab2 2985	. . . . . . . 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 9318	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> x e. CC )
12	4, 11	mulcld 8346	. . . 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 9570	. . . . . 6 |- ( a e. NN -> a e. NN0 )
15		2nn 9466	. . . . . . 7 |- 2 e. NN
16		pw2dvdseu 12946	. . . . . . . 8 |- ( a e. NN -> E! z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) )
17		riotacl 6054	. . . . . . . 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 10989	. . . . . . 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 418	. . . . . 6 |- ( a e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) ) ) e. NN )
21		nn0nndivcl 9629	. . . . . 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 415	. . . . 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 462	. . . . . 6 |- ( ( x e. J /\ y e. NN0 ) <-> ( ( x e. NN /\ -. 2 || x ) /\ y e. NN0 ) )
26	25	anbi1i 462	. . . . 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 12951	. . . . 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 6471	. 2 |- ( T. -> F : ( J X. NN0 ) -1-1-onto-> NN )
31	30	mptru 1411	1 |- F : ( J X. NN0 ) -1-1-onto-> NN

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1402   T. wtru 1403   e. wcel 2209   E!wreu 2530   {crab 2532   class class class wbr 4130   X. cxp 4772   -1-1-onto->wf1o 5376   iota_crio 6037  (class class class)co 6085   e. cmpo 6087   CCcc 8177   RRcr 8178   1c1 8180   + caddc 8182   x. cmul 8184   / cdiv 9002   NNcn 9304   2c2 9355   NN0cn0 9563   ^cexp 10975   || cdvds 12554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-dvds 12555

This theorem is used by:  sqpweven  12953  2sqpwodd  12954  xpnnen  13285