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Theorem negdvdsb 11405
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 9036 . . . 4  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
32anim1i 336 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  e.  ZZ  /\  N  e.  ZZ ) )
4 znegcl 9036 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
54adantl 273 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  -u x  e.  ZZ )
6 zcn 9010 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 9010 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mul2neg 8124 . . . . . . 7  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  -u M )  =  ( x  x.  M ) )
96, 7, 8syl2anr 286 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( -u x  x.  -u M )  =  ( x  x.  M ) )
109adantlr 466 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( -u x  x.  -u M )  =  ( x  x.  M
) )
1110eqeq1d 2124 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( -u x  x.  -u M )  =  N  <->  ( x  x.  M )  =  N ) )
1211biimprd 157 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  -u M )  =  N ) )
131, 3, 5, 12dvds1lem 11400 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  -> 
-u M  ||  N
) )
14 mulneg12 8123 . . . . . . 7  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
156, 7, 14syl2anr 286 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
1615adantlr 466 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
1716eqeq1d 2124 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( -u x  x.  M )  =  N  <->  ( x  x.  -u M )  =  N ) )
1817biimprd 157 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  -u M )  =  N  ->  ( -u x  x.  M )  =  N ) )
193, 1, 5, 18dvds1lem 11400 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  ||  N  ->  M  ||  N
) )
2013, 19impbid 128 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  -u M  ||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   CCcc 7582    x. cmul 7589   -ucneg 7898   ZZcz 9005    || cdvds 11389
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8678  df-z 9006  df-dvds 11390
This theorem is referenced by:  absdvdsb  11407  zdvdsdc  11410  bezoutlemzz  11586  lcmneg  11651
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