Theorem oddpwdclemxy 12337

Description: Lemma for oddpwdc 12342. 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 9152	. . . . . 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 533	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. NN )
4	3	nnzd 9447	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. ZZ )
5		simplr 528	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> Y e. NN0 )
6	2, 5	nnexpcld 10787	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) e. NN )
7	6	nnzd 9447	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) e. ZZ )
8		simpr 110	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> A = ( ( 2 ^ Y ) x. X ) )
9	6, 3	nnmulcld 9039	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( 2 ^ Y ) x. X ) e. NN )
10	8, 9	eqeltrd 2273	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> A e. NN )
11	10	nnzd 9447	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> A e. ZZ )
12	6	nncnd 9004	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) e. CC )
13	3	nncnd 9004	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. CC )
14	12, 13	mulcomd 8048	. . . . . . . . 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 2230	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( X x. ( 2 ^ Y ) ) = A )
16		dvds0lem 11966	. . . . . . . 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 1252	. . . . . . 7 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) || A )
18		simpllr 534	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. 2 || X )
19	8	breq2d 4045	. . . . . . . . . 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 9447	. . . . . . . . . . 11 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> 2 e. ZZ )
21	6	nnne0d 9035	. . . . . . . . . . 11 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) =/= 0 )
22		dvdscmulr 11985	. . . . . . . . . . 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 1253	. . . . . . . . . 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 188	. . . . . . . . 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 674	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. ( ( 2 ^ Y ) x. 2 ) || A )
26	2	nncnd 9004	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> 2 e. CC )
27	26, 5	expp1d 10766	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ ( Y + 1 ) ) = ( ( 2 ^ Y ) x. 2 ) )
28	27	breq1d 4043	. . . . . . . 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 674	. . . . . . 7 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. ( 2 ^ ( Y + 1 ) ) || A )
30		pw2dvdseu 12336	. . . . . . . . 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 5930	. . . . . . . . . . 11 |- ( z = Y -> ( 2 ^ z ) = ( 2 ^ Y ) )
33	32	breq1d 4043	. . . . . . . . . 10 |- ( z = Y -> ( ( 2 ^ z ) || A <-> ( 2 ^ Y ) || A ) )
34		oveq1 5929	. . . . . . . . . . . . 13 |- ( z = Y -> ( z + 1 ) = ( Y + 1 ) )
35	34	oveq2d 5938	. . . . . . . . . . . 12 |- ( z = Y -> ( 2 ^ ( z + 1 ) ) = ( 2 ^ ( Y + 1 ) ) )
36	35	breq1d 4043	. . . . . . . . . . 11 |- ( z = Y -> ( ( 2 ^ ( z + 1 ) ) || A <-> ( 2 ^ ( Y + 1 ) ) || A ) )
37	36	notbid 668	. . . . . . . . . 10 |- ( z = Y -> ( -. ( 2 ^ ( z + 1 ) ) || A <-> -. ( 2 ^ ( Y + 1 ) ) || A ) )
38	33, 37	anbi12d 473	. . . . . . . . 9 |- ( z = Y -> ( ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) <-> ( ( 2 ^ Y ) || A /\ -. ( 2 ^ ( Y + 1 ) ) || A ) ) )
39	38	riota2 5900	. . . . . . . 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 411	. . . . . . 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 945	. . . . . 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 2273	. . . . 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 10787	. . . 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 9004	. . 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 9036	. . 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 2202	. . . . . 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 5938	. . . . 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 5937	. . . 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 2230	. . 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 8869	. 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 306	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 104   <-> wb 105   = wceq 1364   e. wcel 2167   =/= wne 2367   E!wreu 2477   class class class wbr 4033   iota_crio 5876  (class class class)co 5922   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   / cdiv 8699   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ^cexp 10630   || cdvds 11952

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-dvds 11953

This theorem is referenced by:  oddpwdclemdc  12341