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Theorem dvdsnegb 14012
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 10920 . . . 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 10920 . . . 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 10890 . . . . 5  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 10890 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mulneg1 10014 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  -u ( x  x.  M
) )
9 negeq 9831 . . . . . . 7  |-  ( ( x  x.  M )  =  N  ->  -u (
x  x.  M )  =  -u N )
109eqeq2d 2471 . . . . . 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 14006 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  M  ||  -u N
) )
15 zcn 10890 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
16 negeq 9831 . . . . . . . . . 10  |-  ( ( x  x.  M )  =  -u N  ->  -u (
x  x.  M )  =  -u -u N )
17 negneg 9888 . . . . . . . . . 10  |-  ( N  e.  CC  ->  -u -u N  =  N )
1816, 17sylan9eqr 2520 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( x  x.  M
)  =  -u N
)  ->  -u ( x  x.  M )  =  N )
198, 18sylan9eq 2518 . . . . . . . 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 1191 . . . . . 6  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
226, 7, 15, 21syl3an 1270 . . . . 5  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
23223coml 1203 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  M
)  =  -u N  ->  ( -u x  x.  M )  =  N ) )
24233expa 1196 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  = 
-u N  ->  ( -u x  x.  M )  =  N ) )
253, 1, 5, 24dvds1lem 14006 . 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 1395    e. wcel 1819   class class class wbr 4456  (class class class)co 6296   CCcc 9507    x. cmul 9514   -ucneg 9825   ZZcz 10885    || cdvds 13997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-z 10886  df-dvds 13998
This theorem is referenced by:  dvdsabsb  14014  dvdssub  14037  dvdsadd2b  14039  3dvds  14061  bitsfzo  14096  bitscmp  14099  gcdneg  14175  prmdiv  14326  pcneg  14408  znunit  18728  2sqblem  23777  congsym  31068  etransclem9  32187
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