Theorem oddpwdclemodd 12171

Description: Lemma for oddpwdc 12173. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.)

Assertion

Ref	Expression
oddpwdclemodd	|- ( A e. NN -> -. 2 || ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) )

Distinct variable group:   z, A

Proof of Theorem oddpwdclemodd

Step	Hyp	Ref	Expression
1		oddpwdclemndvds 12170	. . 3 |- ( A e. NN -> -. ( 2 ^ ( ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) + 1 ) ) || A )
2		2cn 8989	. . . . 5 |- 2 e. CC
3		pw2dvdseu 12167	. . . . . 6 |- ( A e. NN -> E! z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) )
4		riotacl 5844	. . . . . 6 |- ( E! z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) -> ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) e. NN0 )
5	3, 4	syl 14	. . . . 5 |- ( A e. NN -> ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) e. NN0 )
6		expp1 10526	. . . . 5 |- ( ( 2 e. CC /\ ( 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 ) ) + 1 ) ) = ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) )
7	2, 5, 6	sylancr 414	. . . 4 |- ( A e. NN -> ( 2 ^ ( ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) + 1 ) ) = ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) )
8	7	breq1d 4013	. . 3 |- ( A e. NN -> ( ( 2 ^ ( ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) + 1 ) ) || A <-> ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) || A ) )
9	1, 8	mtbid 672	. 2 |- ( A e. NN -> -. ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) || A )
10		nncn 8926	. . . . . 6 |- ( A e. NN -> A e. CC )
11		2nn 9079	. . . . . . . . 9 |- 2 e. NN
12	11	a1i 9	. . . . . . . 8 |- ( A e. NN -> 2 e. NN )
13	12, 5	nnexpcld 10675	. . . . . . 7 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. NN )
14	13	nncnd 8932	. . . . . 6 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. CC )
15	13	nnap0d 8964	. . . . . 6 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) # 0 )
16	10, 14, 15	divcanap2d 8748	. . . . 5 |- ( A e. NN -> ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) = A )
17	16	eqcomd 2183	. . . 4 |- ( A e. NN -> A = ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) )
18	17	breq2d 4015	. . 3 |- ( A e. NN -> ( ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) || A <-> ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) || ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) ) )
19	12	nnzd 9373	. . . 4 |- ( A e. NN -> 2 e. ZZ )
20		id 19	. . . . . 6 |- ( A e. NN -> A e. NN )
21		oddpwdclemdvds 12169	. . . . . 6 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) || A )
22		nndivdvds 11802	. . . . . . 7 |- ( ( A e. NN /\ ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. NN ) -> ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) || A <-> ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) e. NN ) )
23	22	biimpa 296	. . . . . 6 |- ( ( ( A e. NN /\ ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. NN ) /\ ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) || A ) -> ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) e. NN )
24	20, 13, 21, 23	syl21anc 1237	. . . . 5 |- ( A e. NN -> ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) e. NN )
25	24	nnzd 9373	. . . 4 |- ( A e. NN -> ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) e. ZZ )
26	13	nnzd 9373	. . . 4 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. ZZ )
27	13	nnne0d 8963	. . . 4 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) =/= 0 )
28		dvdscmulr 11826	. . . 4 |- ( ( 2 e. ZZ /\ ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) e. ZZ /\ ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. ZZ /\ ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) =/= 0 ) ) -> ( ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) || ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) <-> 2 || ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) )
29	19, 25, 26, 27, 28	syl112anc 1242	. . 3 |- ( A e. NN -> ( ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) || ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) <-> 2 || ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) )
30	18, 29	bitrd 188	. 2 |- ( A e. NN -> ( ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) || A <-> 2 || ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) )
31	9, 30	mtbid 672	1 |- ( A e. NN -> -. 2 || ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   =/= wne 2347   E!wreu 2457   class class class wbr 4003   iota_crio 5829  (class class class)co 5874   CCcc 7808   0cc0 7810   1c1 7811   + caddc 7813   x. cmul 7815   / cdiv 8628   NNcn 8918   2c2 8969   NN0cn0 9175   ZZcz 9252   ^cexp 10518   || cdvds 11793

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-fz 10008  df-fl 10269  df-mod 10322  df-seqfrec 10445  df-exp 10519  df-dvds 11794

This theorem is referenced by:  oddpwdclemdc  12172