Theorem oddpwdclemxy 11639

Description: Lemma for oddpwdc 11644. Another way of stating that decomposing a natural number into a power of two and an odd number is unique. (Contributed by Jim Kingdon, 16-Nov-2021.)

Assertion

Ref	Expression
oddpwdclemxy	|- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 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 ) ) ) )

Distinct variable groups:   z, A   z, Y

Allowed substitution hint:   X( z)

Proof of Theorem oddpwdclemxy

Step	Hyp	Ref	Expression
1		2nn 8733	. . . . . 6 |- 2 e. NN
2	1	a1i 9	. . . . 5 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> 2 e. NN )
3		simplll 503	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. NN )
4	3	nnzd 9024	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. ZZ )
5		simplr 500	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> Y e. NN0 )
6	2, 5	nnexpcld 10287	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) e. NN )
7	6	nnzd 9024	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) e. ZZ )
8		simpr 109	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> A = ( ( 2 ^ Y ) x. X ) )
9	6, 3	nnmulcld 8627	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( 2 ^ Y ) x. X ) e. NN )
10	8, 9	eqeltrd 2176	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> A e. NN )
11	10	nnzd 9024	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> A e. ZZ )
12	6	nncnd 8592	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) e. CC )
13	3	nncnd 8592	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. CC )
14	12, 13	mulcomd 7659	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( 2 ^ Y ) x. X ) = ( X x. ( 2 ^ Y ) ) )
15	8, 14	eqtr2d 2133	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( X x. ( 2 ^ Y ) ) = A )
16		dvds0lem 11298	. . . . . . . 8 |- ( ( ( X e. ZZ /\ ( 2 ^ Y ) e. ZZ /\ A e. ZZ ) /\ ( X x. ( 2 ^ Y ) ) = A ) -> ( 2 ^ Y ) || A )
17	4, 7, 11, 15, 16	syl31anc 1187	. . . . . . 7 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) || A )
18		simpllr 504	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. 2 || X )
19	8	breq2d 3887	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( ( 2 ^ Y ) x. 2 ) || A <-> ( ( 2 ^ Y ) x. 2 ) || ( ( 2 ^ Y ) x. X ) ) )
20	2	nnzd 9024	. . . . . . . . . . 11 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> 2 e. ZZ )
21	6	nnne0d 8623	. . . . . . . . . . 11 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) =/= 0 )
22		dvdscmulr 11317	. . . . . . . . . . 11 |- ( ( 2 e. ZZ /\ X e. ZZ /\ ( ( 2 ^ Y ) e. ZZ /\ ( 2 ^ Y ) =/= 0 ) ) -> ( ( ( 2 ^ Y ) x. 2 ) || ( ( 2 ^ Y ) x. X ) <-> 2 || X ) )
23	20, 4, 7, 21, 22	syl112anc 1188	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( ( 2 ^ Y ) x. 2 ) || ( ( 2 ^ Y ) x. X ) <-> 2 || X ) )
24	19, 23	bitrd 187	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( ( 2 ^ Y ) x. 2 ) || A <-> 2 || X ) )
25	18, 24	mtbird 639	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. ( ( 2 ^ Y ) x. 2 ) || A )
26	2	nncnd 8592	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> 2 e. CC )
27	26, 5	expp1d 10266	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ ( Y + 1 ) ) = ( ( 2 ^ Y ) x. 2 ) )
28	27	breq1d 3885	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( 2 ^ ( Y + 1 ) ) || A <-> ( ( 2 ^ Y ) x. 2 ) || A ) )
29	25, 28	mtbird 639	. . . . . . 7 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. ( 2 ^ ( Y + 1 ) ) || A )
30		pw2dvdseu 11638	. . . . . . . . 9 |- ( A e. NN -> E! z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) )
31	10, 30	syl 14	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> E! z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) )
32		oveq2 5714	. . . . . . . . . . 11 |- ( z = Y -> ( 2 ^ z ) = ( 2 ^ Y ) )
33	32	breq1d 3885	. . . . . . . . . 10 |- ( z = Y -> ( ( 2 ^ z ) || A <-> ( 2 ^ Y ) || A ) )
34		oveq1 5713	. . . . . . . . . . . . 13 |- ( z = Y -> ( z + 1 ) = ( Y + 1 ) )
35	34	oveq2d 5722	. . . . . . . . . . . 12 |- ( z = Y -> ( 2 ^ ( z + 1 ) ) = ( 2 ^ ( Y + 1 ) ) )
36	35	breq1d 3885	. . . . . . . . . . 11 |- ( z = Y -> ( ( 2 ^ ( z + 1 ) ) || A <-> ( 2 ^ ( Y + 1 ) ) || A ) )
37	36	notbid 633	. . . . . . . . . 10 |- ( z = Y -> ( -. ( 2 ^ ( z + 1 ) ) || A <-> -. ( 2 ^ ( Y + 1 ) ) || A ) )
38	33, 37	anbi12d 460	. . . . . . . . 9 |- ( z = Y -> ( ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) <-> ( ( 2 ^ Y ) || A /\ -. ( 2 ^ ( Y + 1 ) ) || A ) ) )
39	38	riota2 5684	. . . . . . . 8 |- ( ( Y e. NN0 /\ E! z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) -> ( ( ( 2 ^ Y ) || A /\ -. ( 2 ^ ( Y + 1 ) ) || A ) <-> ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) = Y ) )
40	5, 31, 39	syl2anc 406	. . . . . . 7 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( ( 2 ^ Y ) || A /\ -. ( 2 ^ ( Y + 1 ) ) || A ) <-> ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) = Y ) )
41	17, 29, 40	mpbi2and 895	. . . . . 6 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) = Y )
42	41, 5	eqeltrd 2176	. . . . 5 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) e. NN0 )
43	2, 42	nnexpcld 10287	. . . 4 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. NN )
44	43	nncnd 8592	. . 3 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. CC )
45	43	nnap0d 8624	. . 3 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) # 0 )
46	41	eqcomd 2105	. . . . . 6 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> Y = ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) )
47	46	oveq2d 5722	. . . . 5 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) = ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) )
48	47	oveq1d 5721	. . . 4 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( 2 ^ Y ) x. X ) = ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. X ) )
49	8, 48	eqtr2d 2133	. . 3 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. X ) = A )
50	44, 13, 45, 49	mvllmulapd 8459	. 2 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X = ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) )
51	50, 46	jca 302	1 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 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 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   =/= wne 2267   E!wreu 2377   class class class wbr 3875   iota_crio 5661  (class class class)co 5706   0cc0 7500   1c1 7501   + caddc 7503   x. cmul 7505   / cdiv 8293   NNcn 8578   2c2 8629   NN0cn0 8829   ZZcz 8906   ^cexp 10133   || cdvds 11288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-dvds 11289

This theorem is referenced by:  oddpwdclemdc  11643