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Theorem zdiv 10717
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 10359 . . 3  |-  ( M  e.  NN  ->  M  =/=  0 )
21adantr 465 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  M  =/=  0 )
3 nncn 10335 . . 3  |-  ( M  e.  NN  ->  M  e.  CC )
4 zcn 10656 . . 3  |-  ( N  e.  ZZ  ->  N  e.  CC )
5 zcn 10656 . . . . . . . . . . 11  |-  ( k  e.  ZZ  ->  k  e.  CC )
6 divcan3 10023 . . . . . . . . . . . . 13  |-  ( ( k  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  (
( M  x.  k
)  /  M )  =  k )
763coml 1194 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  M  =/=  0  /\  k  e.  CC )  ->  (
( M  x.  k
)  /  M )  =  k )
873expa 1187 . . . . . . . . . . 11  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  k  e.  CC )  ->  ( ( M  x.  k )  /  M )  =  k )
95, 8sylan2 474 . . . . . . . . . 10  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  ->  ( ( M  x.  k )  /  M )  =  k )
1093adantl2 1145 . . . . . . . . 9  |-  ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  ->  ( ( M  x.  k )  /  M
)  =  k )
11 oveq1 6103 . . . . . . . . 9  |-  ( ( M  x.  k )  =  N  ->  (
( M  x.  k
)  /  M )  =  ( N  /  M ) )
1210, 11sylan9req 2496 . . . . . . . 8  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  k  =  ( N  /  M ) )
13 simplr 754 . . . . . . . 8  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  k  e.  ZZ )
1412, 13eqeltrrd 2518 . . . . . . 7  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  ( N  /  M )  e.  ZZ )
1514exp31 604 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  (
k  e.  ZZ  ->  ( ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) ) )
1615rexlimdv 2845 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  ( E. k  e.  ZZ  ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
17 divcan2 10007 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
18173com12 1191 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
19 oveq2 6104 . . . . . . . . 9  |-  ( k  =  ( N  /  M )  ->  ( M  x.  k )  =  ( M  x.  ( N  /  M
) ) )
2019eqeq1d 2451 . . . . . . . 8  |-  ( k  =  ( N  /  M )  ->  (
( M  x.  k
)  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
2120rspcev 3078 . . . . . . 7  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( M  x.  ( N  /  M ) )  =  N )  ->  E. k  e.  ZZ  ( M  x.  k
)  =  N )
2221expcom 435 . . . . . 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 191 . . . 4  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  ( E. k  e.  ZZ  ( M  x.  k
)  =  N  <->  ( N  /  M )  e.  ZZ ) )
25243expia 1189 . . 3  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  =/=  0  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <-> 
( N  /  M
)  e.  ZZ ) ) )
263, 4, 25syl2an 477 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721  (class class class)co 6096   CCcc 9285   0cc0 9287    x. cmul 9292    / cdiv 9998   NNcn 10327   ZZcz 10651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-z 10652
This theorem is referenced by: (None)
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