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Theorem dvdsnegb 12090
Description: An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdsnegb  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  M 
||  -u N ) )

Proof of Theorem dvdsnegb
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 9402 . . . 4  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
32anim2i 342 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  -u N  e.  ZZ ) )
4 znegcl 9402 . . . 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 9376 . . . . 5  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 9376 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mulneg1 8466 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  -u ( x  x.  M
) )
9 negeq 8264 . . . . . . 7  |-  ( ( x  x.  M )  =  N  ->  -u (
x  x.  M )  =  -u N )
109eqeq2d 2216 . . . . . 6  |-  ( ( x  x.  M )  =  N  ->  (
( -u x  x.  M
)  =  -u (
x  x.  M )  <-> 
( -u x  x.  M
)  =  -u N
) )
118, 10syl5ibcom 155 . . . . 5  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
126, 7, 11syl2anr 290 . . . 4  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
1312adantlr 477 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
141, 3, 5, 13dvds1lem 12084 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  M  ||  -u N
) )
15 zcn 9376 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
16 negeq 8264 . . . . . . . . . 10  |-  ( ( x  x.  M )  =  -u N  ->  -u (
x  x.  M )  =  -u -u N )
17 negneg 8321 . . . . . . . . . 10  |-  ( N  e.  CC  ->  -u -u N  =  N )
1816, 17sylan9eqr 2259 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( x  x.  M
)  =  -u N
)  ->  -u ( x  x.  M )  =  N )
198, 18sylan9eq 2257 . . . . . . . 8  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( N  e.  CC  /\  ( x  x.  M )  = 
-u N ) )  ->  ( -u x  x.  M )  =  N )
2019expr 375 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
21203impa 1196 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
226, 7, 15, 21syl3an 1291 . . . . 5  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
23223coml 1212 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
24233expa 1205 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
253, 1, 5, 24dvds1lem 12084 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  -u N  ->  M  ||  N ) )
2614, 25impbid 129 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  M 
||  -u N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1372    e. wcel 2175   class class class wbr 4043  (class class class)co 5943   CCcc 7922    x. cmul 7929   -ucneg 8243   ZZcz 9371    || cdvds 12069
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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-ltadd 8040
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-z 9372  df-dvds 12070
This theorem is referenced by:  dvdsabsb  12092  dvdssub  12120  dvdsadd2b  12122  3dvds  12146  bitscmp  12240  gcdneg  12274  bezoutlemaz  12295  bezoutlembz  12296  prmdiv  12528  pcneg  12619  znunit  14392
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