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Theorem oddpwdclemodd 11777
Description: Lemma for oddpwdc 11779. 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 11776 . . 3  |-  ( A  e.  NN  ->  -.  ( 2 ^ (
( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) )  +  1 ) )  ||  A
)
2 2cn 8759 . . . . 5  |-  2  e.  CC
3 pw2dvdseu 11773 . . . . . 6  |-  ( A  e.  NN  ->  E! z  e.  NN0  ( ( 2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) )
4 riotacl 5712 . . . . . 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 10268 . . . . 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 410 . . . 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 3909 . . 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 646 . 2  |-  ( A  e.  NN  ->  -.  ( ( 2 ^ ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) )  x.  2 )  ||  A
)
10 nncn 8696 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
11 2nn 8849 . . . . . . . . 9  |-  2  e.  NN
1211a1i 9 . . . . . . . 8  |-  ( A  e.  NN  ->  2  e.  NN )
1312, 5nnexpcld 10414 . . . . . . 7  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  e.  NN )
1413nncnd 8702 . . . . . 6  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  e.  CC )
1513nnap0d 8734 . . . . . 6  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) ) #  0 )
1610, 14, 15divcanap2d 8520 . . . . 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 2123 . . . 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 3911 . . 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 9140 . . . 4  |-  ( A  e.  NN  ->  2  e.  ZZ )
20 id 19 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  NN )
21 oddpwdclemdvds 11775 . . . . . 6  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  ||  A
)
22 nndivdvds 11426 . . . . . . 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 1200 . . . . 5  |-  ( A  e.  NN  ->  ( A  /  ( 2 ^ ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) ) )  e.  NN )
2524nnzd 9140 . . . 4  |-  ( A  e.  NN  ->  ( A  /  ( 2 ^ ( iota_ z  e.  NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A
) ) ) )  e.  ZZ )
2613nnzd 9140 . . . 4  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  e.  ZZ )
2713nnne0d 8733 . . . 4  |-  ( A  e.  NN  ->  (
2 ^ ( iota_ z  e.  NN0  ( (
2 ^ z ) 
||  A  /\  -.  ( 2 ^ (
z  +  1 ) )  ||  A ) ) )  =/=  0
)
28 dvdscmulr 11449 . . . 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 1205 . . 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 646 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 1316    e. wcel 1465    =/= wne 2285   E!wreu 2395   class class class wbr 3899   iota_crio 5697  (class class class)co 5742   CCcc 7586   0cc0 7588   1c1 7589    + caddc 7591    x. cmul 7593    / cdiv 8400   NNcn 8688   2c2 8739   NN0cn0 8945   ZZcz 9022   ^cexp 10260    || cdvds 11420
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-dvds 11421
This theorem is referenced by:  oddpwdclemdc  11778
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