MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvdsnegb Unicode version

Theorem dvdsnegb 12796
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 20 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 znegcl 10247 . . . 4  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
32anim2i 553 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  -u N  e.  ZZ ) )
4 znegcl 10247 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
54adantl 453 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  -u x  e.  ZZ )
6 zcn 10221 . . . . 5  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 10221 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mulneg1 9404 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  -u ( x  x.  M
) )
9 negeq 9232 . . . . . . 7  |-  ( ( x  x.  M )  =  N  ->  -u (
x  x.  M )  =  -u N )
109eqeq2d 2400 . . . . . 6  |-  ( ( x  x.  M )  =  N  ->  (
( -u x  x.  M
)  =  -u (
x  x.  M )  <-> 
( -u x  x.  M
)  =  -u N
) )
118, 10syl5ibcom 212 . . . . 5  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
126, 7, 11syl2anr 465 . . . 4  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
1312adantlr 696 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  M )  =  -u N ) )
141, 3, 5, 13dvds1lem 12790 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  M  ||  -u N
) )
15 zcn 10221 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
16 negeq 9232 . . . . . . . . . 10  |-  ( ( x  x.  M )  =  -u N  ->  -u (
x  x.  M )  =  -u -u N )
17 negneg 9285 . . . . . . . . . 10  |-  ( N  e.  CC  ->  -u -u N  =  N )
1816, 17sylan9eqr 2443 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( x  x.  M
)  =  -u N
)  ->  -u ( x  x.  M )  =  N )
198, 18sylan9eq 2441 . . . . . . . 8  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( N  e.  CC  /\  ( x  x.  M )  = 
-u N ) )  ->  ( -u x  x.  M )  =  N )
2019expr 599 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
21203impa 1148 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
226, 7, 15, 21syl3an 1226 . . . . 5  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
23223coml 1160 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
24233expa 1153 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
253, 1, 5, 24dvds1lem 12790 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  -u N  ->  M  ||  N ) )
2614, 25impbid 184 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  M 
||  -u N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4155  (class class class)co 6022   CCcc 8923    x. cmul 8930   -ucneg 9226   ZZcz 10216    || cdivides 12781
This theorem is referenced by:  dvdsabsb  12798  dvdssub  12819  dvdsadd2b  12821  3dvds  12841  bitsfzo  12876  bitscmp  12879  gcdneg  12955  prmdiv  13103  pcneg  13176  znunit  16769  2sqblem  21030  congsym  26726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-z 10217  df-dvds 12782
  Copyright terms: Public domain W3C validator