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Theorem dvdsnegb 13550
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 22 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 znegcl 10680 . . . 4  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
32anim2i 569 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  -u N  e.  ZZ ) )
4 znegcl 10680 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
54adantl 466 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  -u x  e.  ZZ )
6 zcn 10651 . . . . 5  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 10651 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mulneg1 9781 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  -u ( x  x.  M
) )
9 negeq 9602 . . . . . . 7  |-  ( ( x  x.  M )  =  N  ->  -u (
x  x.  M )  =  -u N )
109eqeq2d 2454 . . . . . 6  |-  ( ( x  x.  M )  =  N  ->  (
( -u x  x.  M
)  =  -u (
x  x.  M )  <-> 
( -u x  x.  M
)  =  -u N
) )
118, 10syl5ibcom 220 . . . . 5  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
126, 7, 11syl2anr 478 . . . 4  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
1312adantlr 714 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
141, 3, 5, 13dvds1lem 13544 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  M  ||  -u N
) )
15 zcn 10651 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
16 negeq 9602 . . . . . . . . . 10  |-  ( ( x  x.  M )  =  -u N  ->  -u (
x  x.  M )  =  -u -u N )
17 negneg 9659 . . . . . . . . . 10  |-  ( N  e.  CC  ->  -u -u N  =  N )
1816, 17sylan9eqr 2497 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( x  x.  M
)  =  -u N
)  ->  -u ( x  x.  M )  =  N )
198, 18sylan9eq 2495 . . . . . . . 8  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( N  e.  CC  /\  ( x  x.  M )  = 
-u N ) )  ->  ( -u x  x.  M )  =  N )
2019expr 615 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
21203impa 1182 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
226, 7, 15, 21syl3an 1260 . . . . 5  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
23223coml 1194 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
24233expa 1187 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
253, 1, 5, 24dvds1lem 13544 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  -u N  ->  M  ||  N ) )
2614, 25impbid 191 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  M 
||  -u N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4292  (class class class)co 6091   CCcc 9280    x. cmul 9287   -ucneg 9596   ZZcz 10646    || cdivides 13535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-z 10647  df-dvds 13536
This theorem is referenced by:  dvdsabsb  13552  dvdssub  13573  dvdsadd2b  13575  3dvds  13596  bitsfzo  13631  bitscmp  13634  gcdneg  13710  prmdiv  13860  pcneg  13940  znunit  17996  2sqblem  22716  congsym  29311
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