Theorem oddpwdclemxy 12123

Description: Lemma for oddpwdc 12128. 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 9039	. . . . . 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 528	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. NN )
4	3	nnzd 9333	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. ZZ )
5		simplr 525	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> Y e. NN0 )
6	2, 5	nnexpcld 10631	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) e. NN )
7	6	nnzd 9333	. . . . . . . 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 8927	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( ( 2 ^ Y ) x. X ) e. NN )
10	8, 9	eqeltrd 2247	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> A e. NN )
11	10	nnzd 9333	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> A e. ZZ )
12	6	nncnd 8892	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) e. CC )
13	3	nncnd 8892	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> X e. CC )
14	12, 13	mulcomd 7941	. . . . . . . . 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 2204	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( X x. ( 2 ^ Y ) ) = A )
16		dvds0lem 11763	. . . . . . . 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 1236	. . . . . . 7 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) || A )
18		simpllr 529	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. 2 || X )
19	8	breq2d 4001	. . . . . . . . . 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 9333	. . . . . . . . . . 11 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> 2 e. ZZ )
21	6	nnne0d 8923	. . . . . . . . . . 11 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ Y ) =/= 0 )
22		dvdscmulr 11782	. . . . . . . . . . 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 1237	. . . . . . . . . 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 668	. . . . . . . 8 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. ( ( 2 ^ Y ) x. 2 ) || A )
26	2	nncnd 8892	. . . . . . . . . 10 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> 2 e. CC )
27	26, 5	expp1d 10610	. . . . . . . . 9 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( 2 ^ ( Y + 1 ) ) = ( ( 2 ^ Y ) x. 2 ) )
28	27	breq1d 3999	. . . . . . . 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 668	. . . . . . 7 |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> -. ( 2 ^ ( Y + 1 ) ) || A )
30		pw2dvdseu 12122	. . . . . . . . 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 5861	. . . . . . . . . . 11 |- ( z = Y -> ( 2 ^ z ) = ( 2 ^ Y ) )
33	32	breq1d 3999	. . . . . . . . . 10 |- ( z = Y -> ( ( 2 ^ z ) || A <-> ( 2 ^ Y ) || A ) )
34		oveq1 5860	. . . . . . . . . . . . 13 |- ( z = Y -> ( z + 1 ) = ( Y + 1 ) )
35	34	oveq2d 5869	. . . . . . . . . . . 12 |- ( z = Y -> ( 2 ^ ( z + 1 ) ) = ( 2 ^ ( Y + 1 ) ) )
36	35	breq1d 3999	. . . . . . . . . . 11 |- ( z = Y -> ( ( 2 ^ ( z + 1 ) ) || A <-> ( 2 ^ ( Y + 1 ) ) || A ) )
37	36	notbid 662	. . . . . . . . . 10 |- ( z = Y -> ( -. ( 2 ^ ( z + 1 ) ) || A <-> -. ( 2 ^ ( Y + 1 ) ) || A ) )
38	33, 37	anbi12d 470	. . . . . . . . 9 |- ( z = Y -> ( ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) <-> ( ( 2 ^ Y ) || A /\ -. ( 2 ^ ( Y + 1 ) ) || A ) ) )
39	38	riota2 5831	. . . . . . . 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 409	. . . . . . 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 938	. . . . . 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 2247	. . . . 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 10631	. . . 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 8892	. . 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 8924	. . 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 2176	. . . . . 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 5869	. . . . 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 5868	. . . 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 2204	. . 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 8759	. 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 304	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 1348   e. wcel 2141   =/= wne 2340   E!wreu 2450   class class class wbr 3989   iota_crio 5808  (class class class)co 5853   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   / cdiv 8589   NNcn 8878   2c2 8929   NN0cn0 9135   ZZcz 9212   ^cexp 10475   || cdvds 11749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-dvds 11750

This theorem is referenced by:  oddpwdclemdc  12127