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Theorem zdiv 9258
Description: Two ways to express " M divides  N. (Contributed by NM, 3-Oct-2008.)
Assertion
Ref Expression
zdiv  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <-> 
( N  /  M
)  e.  ZZ ) )
Distinct variable groups:    k, M    k, N

Proof of Theorem zdiv
StepHypRef Expression
1 nnap0 8868 . . 3  |-  ( M  e.  NN  ->  M #  0 )
21adantr 274 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  M #  0 )
3 nncn 8847 . . 3  |-  ( M  e.  NN  ->  M  e.  CC )
4 zcn 9178 . . 3  |-  ( N  e.  ZZ  ->  N  e.  CC )
5 zcn 9178 . . . . . . . . . . 11  |-  ( k  e.  ZZ  ->  k  e.  CC )
6 divcanap3 8576 . . . . . . . . . . . . 13  |-  ( ( k  e.  CC  /\  M  e.  CC  /\  M #  0 )  ->  (
( M  x.  k
)  /  M )  =  k )
763coml 1192 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  M #  0  /\  k  e.  CC )  ->  (
( M  x.  k
)  /  M )  =  k )
873expa 1185 . . . . . . . . . . 11  |-  ( ( ( M  e.  CC  /\  M #  0 )  /\  k  e.  CC )  ->  ( ( M  x.  k )  /  M
)  =  k )
95, 8sylan2 284 . . . . . . . . . 10  |-  ( ( ( M  e.  CC  /\  M #  0 )  /\  k  e.  ZZ )  ->  ( ( M  x.  k )  /  M
)  =  k )
1093adantl2 1139 . . . . . . . . 9  |-  ( ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  /\  k  e.  ZZ )  ->  (
( M  x.  k
)  /  M )  =  k )
11 oveq1 5834 . . . . . . . . 9  |-  ( ( M  x.  k )  =  N  ->  (
( M  x.  k
)  /  M )  =  ( N  /  M ) )
1210, 11sylan9req 2211 . . . . . . . 8  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  k  =  ( N  /  M ) )
13 simplr 520 . . . . . . . 8  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  k  e.  ZZ )
1412, 13eqeltrrd 2235 . . . . . . 7  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  ( N  /  M )  e.  ZZ )
1514exp31 362 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  ->  (
k  e.  ZZ  ->  ( ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) ) )
1615rexlimdv 2573 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  ->  ( E. k  e.  ZZ  ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
17 divcanap2 8558 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC  /\  M #  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
18173com12 1189 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
19 oveq2 5835 . . . . . . . . 9  |-  ( k  =  ( N  /  M )  ->  ( M  x.  k )  =  ( M  x.  ( N  /  M
) ) )
2019eqeq1d 2166 . . . . . . . 8  |-  ( k  =  ( N  /  M )  ->  (
( M  x.  k
)  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
2120rspcev 2816 . . . . . . 7  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( M  x.  ( N  /  M ) )  =  N )  ->  E. k  e.  ZZ  ( M  x.  k
)  =  N )
2221expcom 115 . . . . . 6  |-  ( ( M  x.  ( N  /  M ) )  =  N  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  ( M  x.  k )  =  N ) )
2318, 22syl 14 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  ( M  x.  k )  =  N ) )
2416, 23impbid 128 . . . 4  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  ->  ( E. k  e.  ZZ  ( M  x.  k
)  =  N  <->  ( N  /  M )  e.  ZZ ) )
25243expia 1187 . . 3  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M #  0  -> 
( E. k  e.  ZZ  ( M  x.  k )  =  N  <-> 
( N  /  M
)  e.  ZZ ) ) )
263, 4, 25syl2an 287 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M #  0  -> 
( E. k  e.  ZZ  ( M  x.  k )  =  N  <-> 
( N  /  M
)  e.  ZZ ) ) )
272, 26mpd 13 1  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <-> 
( N  /  M
)  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   E.wrex 2436   class class class wbr 3967  (class class class)co 5827   CCcc 7733   0cc0 7735    x. cmul 7740   # cap 8461    / cdiv 8550   NNcn 8839   ZZcz 9173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4029  df-id 4256  df-po 4259  df-iso 4260  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-iota 5138  df-fun 5175  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-z 9174
This theorem is referenced by:  addmodlteq  10307
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