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Theorem negdvdsb 14084
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 22 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 znegcl 10895 . . . 4  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
32anim1i 566 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  e.  ZZ  /\  N  e.  ZZ ) )
4 znegcl 10895 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
54adantl 464 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  -u x  e.  ZZ )
6 zcn 10865 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 10865 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mul2neg 9992 . . . . . . 7  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  -u M )  =  ( x  x.  M ) )
96, 7, 8syl2anr 476 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( -u x  x.  -u M )  =  ( x  x.  M ) )
109adantlr 712 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( -u x  x.  -u M )  =  ( x  x.  M
) )
1110eqeq1d 2456 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( -u x  x.  -u M )  =  N  <->  ( x  x.  M )  =  N ) )
1211biimprd 223 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  -u M )  =  N ) )
131, 3, 5, 12dvds1lem 14079 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  -> 
-u M  ||  N
) )
14 mulneg12 9991 . . . . . . 7  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
156, 7, 14syl2anr 476 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
1615adantlr 712 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
1716eqeq1d 2456 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( -u x  x.  M )  =  N  <->  ( x  x.  -u M )  =  N ) )
1817biimprd 223 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  -u M )  =  N  ->  ( -u x  x.  M )  =  N ) )
193, 1, 5, 18dvds1lem 14079 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  ||  N  ->  M  ||  N
) )
2013, 19impbid 191 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  -u M  ||  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   class class class wbr 4439  (class class class)co 6270   CCcc 9479    x. cmul 9486   -ucneg 9797   ZZcz 10860    || cdvds 14070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-z 10861  df-dvds 14071
This theorem is referenced by:  absdvdsb  14086  3dvds  14134  lcmneg  31450
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