Theorem oddpwdc 12745

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 9215	. . . . . 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 10934	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> ( 2 ^ y ) e. CC )
5		breq2 4092	. . . . . . . . . 10 |- ( z = x -> ( 2 || z <-> 2 || x ) )
6	5	notbid 673	. . . . . . . . 9 |- ( z = x -> ( -. 2 || z <-> -. 2 || x ) )
7		oddpwdc.j	. . . . . . . . 9 |- J = { z e. NN | -. 2 || z }
8	6, 7	elrab2 2965	. . . . . . . 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 9156	. . . . 5 |- ( ( x e. J /\ y e. NN0 ) -> x e. CC )
12	4, 11	mulcld 8199	. . . 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 9408	. . . . . 6 |- ( a e. NN -> a e. NN0 )
15		2nn 9304	. . . . . . 7 |- 2 e. NN
16		pw2dvdseu 12739	. . . . . . . 8 |- ( a e. NN -> E! z e. NN0 ( ( 2 ^ z ) || a /\ -. ( 2 ^ ( z + 1 ) ) || a ) )
17		riotacl 5986	. . . . . . . 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 10813	. . . . . . 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 9463	. . . . . 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 12744	. . . . 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 6399	. 2 |- ( T. -> F : ( J X. NN0 ) -1-1-onto-> NN )
31	30	mptru 1406	1 |- F : ( J X. NN0 ) -1-1-onto-> NN

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1397   T. wtru 1398   e. wcel 2202   E!wreu 2512   {crab 2514   class class class wbr 4088   X. cxp 4723   -1-1-onto->wf1o 5325   iota_crio 5969  (class class class)co 6017   e. cmpo 6019   CCcc 8029   RRcr 8030   1c1 8032   + caddc 8034   x. cmul 8036   / cdiv 8851   NNcn 9142   2c2 9193   NN0cn0 9401   ^cexp 10799   || cdvds 12347

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-dvds 12348

This theorem is referenced by:  sqpweven  12746  2sqpwodd  12747  xpnnen  13014