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Theorem negdvdsb 11841
Description: An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
negdvdsb  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  -u M  ||  N ) )

Proof of Theorem negdvdsb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 19 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 znegcl 9309 . . . 4  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
32anim1i 340 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  e.  ZZ  /\  N  e.  ZZ ) )
4 znegcl 9309 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
54adantl 277 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  -u x  e.  ZZ )
6 zcn 9283 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 9283 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mul2neg 8380 . . . . . . 7  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  -u M )  =  ( x  x.  M ) )
96, 7, 8syl2anr 290 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( -u x  x.  -u M )  =  ( x  x.  M ) )
109adantlr 477 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( -u x  x.  -u M )  =  ( x  x.  M
) )
1110eqeq1d 2198 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( -u x  x.  -u M )  =  N  <->  ( x  x.  M )  =  N ) )
1211biimprd 158 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  -u M )  =  N ) )
131, 3, 5, 12dvds1lem 11836 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  -> 
-u M  ||  N
) )
14 mulneg12 8379 . . . . . . 7  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
156, 7, 14syl2anr 290 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
1615adantlr 477 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
1716eqeq1d 2198 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( -u x  x.  M )  =  N  <->  ( x  x.  -u M )  =  N ) )
1817biimprd 158 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  -u M )  =  N  ->  ( -u x  x.  M )  =  N ) )
193, 1, 5, 18dvds1lem 11836 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  ||  N  ->  M  ||  N
) )
2013, 19impbid 129 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  -u M  ||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5892   CCcc 7834    x. cmul 7841   -ucneg 8154   ZZcz 9278    || cdvds 11821
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-ltadd 7952
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5234  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-inn 8945  df-z 9279  df-dvds 11822
This theorem is referenced by:  absdvdsb  11843  zdvdsdc  11846  bezoutlemzz  12030  lcmneg  12101
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