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Theorem zdiv 10296
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 nnne0 9988 . . 3  |-  ( M  e.  NN  ->  M  =/=  0 )
21adantr 452 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  M  =/=  0 )
3 nncn 9964 . . 3  |-  ( M  e.  NN  ->  M  e.  CC )
4 zcn 10243 . . 3  |-  ( N  e.  ZZ  ->  N  e.  CC )
5 zcn 10243 . . . . . . . . . . 11  |-  ( k  e.  ZZ  ->  k  e.  CC )
6 divcan3 9658 . . . . . . . . . . . . 13  |-  ( ( k  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  (
( M  x.  k
)  /  M )  =  k )
763coml 1160 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  M  =/=  0  /\  k  e.  CC )  ->  (
( M  x.  k
)  /  M )  =  k )
873expa 1153 . . . . . . . . . . 11  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  k  e.  CC )  ->  ( ( M  x.  k )  /  M )  =  k )
95, 8sylan2 461 . . . . . . . . . 10  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  ->  ( ( M  x.  k )  /  M )  =  k )
1093adantl2 1114 . . . . . . . . 9  |-  ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  ->  ( ( M  x.  k )  /  M
)  =  k )
11 oveq1 6047 . . . . . . . . 9  |-  ( ( M  x.  k )  =  N  ->  (
( M  x.  k
)  /  M )  =  ( N  /  M ) )
1210, 11sylan9req 2457 . . . . . . . 8  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  k  =  ( N  /  M ) )
13 simplr 732 . . . . . . . 8  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  k  e.  ZZ )
1412, 13eqeltrrd 2479 . . . . . . 7  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  ( N  /  M )  e.  ZZ )
1514exp31 588 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  (
k  e.  ZZ  ->  ( ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) ) )
1615rexlimdv 2789 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  ( E. k  e.  ZZ  ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
17 divcan2 9642 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
18173com12 1157 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
19 oveq2 6048 . . . . . . . . 9  |-  ( k  =  ( N  /  M )  ->  ( M  x.  k )  =  ( M  x.  ( N  /  M
) ) )
2019eqeq1d 2412 . . . . . . . 8  |-  ( k  =  ( N  /  M )  ->  (
( M  x.  k
)  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
2120rspcev 3012 . . . . . . 7  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( M  x.  ( N  /  M ) )  =  N )  ->  E. k  e.  ZZ  ( M  x.  k
)  =  N )
2221expcom 425 . . . . . 6  |-  ( ( M  x.  ( N  /  M ) )  =  N  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  ( M  x.  k )  =  N ) )
2318, 22syl 16 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  ( M  x.  k )  =  N ) )
2416, 23impbid 184 . . . 4  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  ( E. k  e.  ZZ  ( M  x.  k
)  =  N  <->  ( N  /  M )  e.  ZZ ) )
25243expia 1155 . . 3  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  =/=  0  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <-> 
( N  /  M
)  e.  ZZ ) ) )
263, 4, 25syl2an 464 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  =/=  0  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <-> 
( N  /  M
)  e.  ZZ ) ) )
272, 26mpd 15 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667  (class class class)co 6040   CCcc 8944   0cc0 8946    x. cmul 8951    / cdiv 9633   NNcn 9956   ZZcz 10238
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-z 10239
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