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Theorem zdiv 10894
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 10529 . . 3  |-  ( M  e.  NN  ->  M  =/=  0 )
21adantr 463 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  M  =/=  0 )
3 nncn 10504 . . 3  |-  ( M  e.  NN  ->  M  e.  CC )
4 zcn 10830 . . 3  |-  ( N  e.  ZZ  ->  N  e.  CC )
5 zcn 10830 . . . . . . . . . . 11  |-  ( k  e.  ZZ  ->  k  e.  CC )
6 divcan3 10192 . . . . . . . . . . . . 13  |-  ( ( k  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  (
( M  x.  k
)  /  M )  =  k )
763coml 1204 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  M  =/=  0  /\  k  e.  CC )  ->  (
( M  x.  k
)  /  M )  =  k )
873expa 1197 . . . . . . . . . . 11  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  k  e.  CC )  ->  ( ( M  x.  k )  /  M )  =  k )
95, 8sylan2 472 . . . . . . . . . 10  |-  ( ( ( M  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  ->  ( ( M  x.  k )  /  M )  =  k )
1093adantl2 1154 . . . . . . . . 9  |-  ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  ->  ( ( M  x.  k )  /  M
)  =  k )
11 oveq1 6241 . . . . . . . . 9  |-  ( ( M  x.  k )  =  N  ->  (
( M  x.  k
)  /  M )  =  ( N  /  M ) )
1210, 11sylan9req 2464 . . . . . . . 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 2491 . . . . . . 7  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  ( N  /  M )  e.  ZZ )
1514exp31 602 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  (
k  e.  ZZ  ->  ( ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) ) )
1615rexlimdv 2893 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  ( E. k  e.  ZZ  ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
17 divcan2 10176 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
18173com12 1201 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M  =/=  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
19 oveq2 6242 . . . . . . . . 9  |-  ( k  =  ( N  /  M )  ->  ( M  x.  k )  =  ( M  x.  ( N  /  M
) ) )
2019eqeq1d 2404 . . . . . . . 8  |-  ( k  =  ( N  /  M )  ->  (
( M  x.  k
)  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
2120rspcev 3159 . . . . . . 7  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( M  x.  ( N  /  M ) )  =  N )  ->  E. k  e.  ZZ  ( M  x.  k
)  =  N )
2221expcom 433 . . . . . 6  |-  ( ( M  x.  ( N  /  M ) )  =  N  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  ( M  x.  k )  =  N ) )
2318, 22syl 17 . . . . 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 1199 . . 3  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  =/=  0  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <-> 
( N  /  M
)  e.  ZZ ) ) )
263, 4, 25syl2an 475 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754  (class class class)co 6234   CCcc 9440   0cc0 9442    x. cmul 9447    / cdiv 10167   NNcn 10496   ZZcz 10825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-z 10826
This theorem is referenced by: (None)
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