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Theorem negdvdsb 12636
Description: An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
negdvdsb  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  -u M  ||  N ) )

Proof of Theorem negdvdsb
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 10144 . . . 4  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
32anim1i 551 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  e.  ZZ  /\  N  e.  ZZ ) )
4 znegcl 10144 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
54adantl 452 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  -u x  e.  ZZ )
6 zcn 10118 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 10118 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 mul2neg 9306 . . . . . . 7  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  -u M )  =  ( x  x.  M ) )
96, 7, 8syl2anr 464 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( -u x  x.  -u M )  =  ( x  x.  M ) )
109adantlr 695 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( -u x  x.  -u M )  =  ( x  x.  M
) )
1110eqeq1d 2366 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( -u x  x.  -u M )  =  N  <->  ( x  x.  M )  =  N ) )
1211biimprd 214 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( -u x  x.  -u M )  =  N ) )
131, 3, 5, 12dvds1lem 12631 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  -> 
-u M  ||  N
) )
14 mulneg12 9305 . . . . . . 7  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
156, 7, 14syl2anr 464 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
1615adantlr 695 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( -u x  x.  M )  =  ( x  x.  -u M
) )
1716eqeq1d 2366 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( -u x  x.  M )  =  N  <->  ( x  x.  -u M )  =  N ) )
1817biimprd 214 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  -u M )  =  N  ->  ( -u x  x.  M )  =  N ) )
193, 1, 5, 18dvds1lem 12631 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  ||  N  ->  M  ||  N
) )
2013, 19impbid 183 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  -u M  ||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   class class class wbr 4102  (class class class)co 5942   CCcc 8822    x. cmul 8829   -ucneg 9125   ZZcz 10113    || cdivides 12622
This theorem is referenced by:  absdvdsb  12638  3dvds  12682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-z 10114  df-dvds 12623
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