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Theorem dvdsnegb 10719
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 8717 . . . 4  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
32anim2i 334 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  -u N  e.  ZZ ) )
4 znegcl 8717 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
54adantl 271 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  -u x  e.  ZZ )
6 zcn 8691 . . . . 5  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 8691 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mulneg1 7820 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  -u ( x  x.  M
) )
9 negeq 7622 . . . . . . 7  |-  ( ( x  x.  M )  =  N  ->  -u (
x  x.  M )  =  -u N )
109eqeq2d 2096 . . . . . 6  |-  ( ( x  x.  M )  =  N  ->  (
( -u x  x.  M
)  =  -u (
x  x.  M )  <-> 
( -u x  x.  M
)  =  -u N
) )
118, 10syl5ibcom 153 . . . . 5  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
126, 7, 11syl2anr 284 . . . 4  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
1312adantlr 461 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
141, 3, 5, 13dvds1lem 10713 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  M  ||  -u N
) )
15 zcn 8691 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
16 negeq 7622 . . . . . . . . . 10  |-  ( ( x  x.  M )  =  -u N  ->  -u (
x  x.  M )  =  -u -u N )
17 negneg 7679 . . . . . . . . . 10  |-  ( N  e.  CC  ->  -u -u N  =  N )
1816, 17sylan9eqr 2139 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( x  x.  M
)  =  -u N
)  ->  -u ( x  x.  M )  =  N )
198, 18sylan9eq 2137 . . . . . . . 8  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( N  e.  CC  /\  ( x  x.  M )  = 
-u N ) )  ->  ( -u x  x.  M )  =  N )
2019expr 367 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
21203impa 1136 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
226, 7, 15, 21syl3an 1214 . . . . 5  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
23223coml 1148 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
24233expa 1141 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
253, 1, 5, 24dvds1lem 10713 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  -u N  ->  M  ||  N ) )
2614, 25impbid 127 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  M 
||  -u N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   class class class wbr 3822  (class class class)co 5615   CCcc 7295    x. cmul 7302   -ucneg 7601   ZZcz 8686    || cdvds 10702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-1re 7386  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-addcom 7392  ax-mulcom 7393  ax-addass 7394  ax-distr 7396  ax-i2m1 7397  ax-0lt1 7398  ax-0id 7400  ax-rnegex 7401  ax-cnre 7403  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406  ax-pre-ltadd 7408
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-iota 4948  df-fun 4985  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-sub 7602  df-neg 7603  df-inn 8361  df-z 8687  df-dvds 10703
This theorem is referenced by:  dvdsabsb  10721  dvdssub  10747  dvdsadd2b  10749  gcdneg  10879  bezoutlemaz  10898  bezoutlembz  10899
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