Theorem oddpwdclemodd 12046

Description: Lemma for oddpwdc 12048. 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 12045	. . 3 |- ( A e. NN -> -. ( 2 ^ ( ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) + 1 ) ) || A )
2		2cn 8904	. . . . 5 |- 2 e. CC
3		pw2dvdseu 12042	. . . . . 6 |- ( A e. NN -> E! z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) )
4		riotacl 5794	. . . . . 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 10426	. . . . 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 411	. . . 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 3975	. . 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 662	. 2 |- ( A e. NN -> -. ( ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) x. 2 ) || A )
10		nncn 8841	. . . . . 6 |- ( A e. NN -> A e. CC )
11		2nn 8994	. . . . . . . . 9 |- 2 e. NN
12	11	a1i 9	. . . . . . . 8 |- ( A e. NN -> 2 e. NN )
13	12, 5	nnexpcld 10573	. . . . . . 7 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. NN )
14	13	nncnd 8847	. . . . . 6 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. CC )
15	13	nnap0d 8879	. . . . . 6 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) # 0 )
16	10, 14, 15	divcanap2d 8665	. . . . 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 2163	. . . 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 3977	. . 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 9285	. . . 4 |- ( A e. NN -> 2 e. ZZ )
20		id 19	. . . . . 6 |- ( A e. NN -> A e. NN )
21		oddpwdclemdvds 12044	. . . . . 6 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) || A )
22		nndivdvds 11692	. . . . . . 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 294	. . . . . 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 1219	. . . . 5 |- ( A e. NN -> ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) e. NN )
25	24	nnzd 9285	. . . 4 |- ( A e. NN -> ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) e. ZZ )
26	13	nnzd 9285	. . . 4 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) e. ZZ )
27	13	nnne0d 8878	. . . 4 |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) =/= 0 )
28		dvdscmulr 11715	. . . 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 1224	. . 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 187	. 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 662	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 103   <-> wb 104   = wceq 1335   e. wcel 2128   =/= wne 2327   E!wreu 2437   class class class wbr 3965   iota_crio 5779  (class class class)co 5824   CCcc 7730   0cc0 7732   1c1 7733   + caddc 7735   x. cmul 7737   / cdiv 8545   NNcn 8833   2c2 8884   NN0cn0 9090   ZZcz 9167   ^cexp 10418   || cdvds 11683

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fl 10169  df-mod 10222  df-seqfrec 10345  df-exp 10419  df-dvds 11684

This theorem is referenced by:  oddpwdclemdc  12047