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Theorem zdiv 9300
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 8907 . . 3  |-  ( M  e.  NN  ->  M #  0 )
21adantr 274 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  M #  0 )
3 nncn 8886 . . 3  |-  ( M  e.  NN  ->  M  e.  CC )
4 zcn 9217 . . 3  |-  ( N  e.  ZZ  ->  N  e.  CC )
5 zcn 9217 . . . . . . . . . . 11  |-  ( k  e.  ZZ  ->  k  e.  CC )
6 divcanap3 8615 . . . . . . . . . . . . 13  |-  ( ( k  e.  CC  /\  M  e.  CC  /\  M #  0 )  ->  (
( M  x.  k
)  /  M )  =  k )
763coml 1205 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  M #  0  /\  k  e.  CC )  ->  (
( M  x.  k
)  /  M )  =  k )
873expa 1198 . . . . . . . . . . 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 1149 . . . . . . . . 9  |-  ( ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  /\  k  e.  ZZ )  ->  (
( M  x.  k
)  /  M )  =  k )
11 oveq1 5860 . . . . . . . . 9  |-  ( ( M  x.  k )  =  N  ->  (
( M  x.  k
)  /  M )  =  ( N  /  M ) )
1210, 11sylan9req 2224 . . . . . . . 8  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  k  =  ( N  /  M ) )
13 simplr 525 . . . . . . . 8  |-  ( ( ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  /\  k  e.  ZZ )  /\  ( M  x.  k )  =  N )  ->  k  e.  ZZ )
1412, 13eqeltrrd 2248 . . . . . . 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 2586 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  ->  ( E. k  e.  ZZ  ( M  x.  k
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
17 divcanap2 8597 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC  /\  M #  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
18173com12 1202 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  M #  0 )  ->  ( M  x.  ( N  /  M ) )  =  N )
19 oveq2 5861 . . . . . . . . 9  |-  ( k  =  ( N  /  M )  ->  ( M  x.  k )  =  ( M  x.  ( N  /  M
) ) )
2019eqeq1d 2179 . . . . . . . 8  |-  ( k  =  ( N  /  M )  ->  (
( M  x.  k
)  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
2120rspcev 2834 . . . . . . 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 1200 . . 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 973    = wceq 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774    x. cmul 7779   # cap 8500    / cdiv 8589   NNcn 8878   ZZcz 9212
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-z 9213
This theorem is referenced by:  addmodlteq  10354
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