ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oddpwdclemodd Unicode version

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
StepHypRef 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 )
53, 4syl 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 ) )
72, 5, 6sylancr 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 ) )
87breq1d 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 ) )
91, 8mtbid 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
1211a1i 9 . . . . . . . 8  |-  ( A  e.  NN  ->  2  e.  NN )
1312, 5nnexpcld 10573 . . . . . . 7  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  e.  NN )
1413nncnd 8847 . . . . . 6  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  e.  CC )
1513nnap0d 8879 . . . . . 6  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) ) #  0 )
1610, 14, 15divcanap2d 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 )
1716eqcomd 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 ) ) ) ) ) )
1817breq2d 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
) ) ) ) ) ) )
1912nnzd 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 ) )
2322biimpa 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 )
2420, 13, 21, 23syl21anc 1219 . . . . 5  |-  ( A  e.  NN  ->  ( A  /  ( 2 ^ ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) ) )  e.  NN )
2524nnzd 9285 . . . 4  |-  ( A  e.  NN  ->  ( A  /  ( 2 ^ ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) ) )  e.  ZZ )
2613nnzd 9285 . . . 4  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  e.  ZZ )
2713nnne0d 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
) ) ) ) ) )
2919, 25, 26, 27, 28syl112anc 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
) ) ) ) ) )
3018, 29bitrd 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 ) ) ) ) ) )
319, 30mtbid 662 1  |-  ( A  e.  NN  ->  -.  2  ||  ( A  / 
( 2 ^ ( iota_ z  e.  NN0  (
( 2 ^ z
)  ||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) ) ) )
Colors of variables: wff set class
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
  Copyright terms: Public domain W3C validator