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Theorem dvdsval2 11423
Description: One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
Assertion
Ref Expression
dvdsval2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )

Proof of Theorem dvdsval2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 divides 11422 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
213adant2 985 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  ( k  x.  M )  =  N ) )
3 zcn 9027 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
433ad2ant3 989 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  N  e.  CC )
54adantr 274 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  N  e.  CC )
6 zcn 9027 . . . . . . . . . 10  |-  ( k  e.  ZZ  ->  k  e.  CC )
76adantl 275 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  k  e.  CC )
8 zcn 9027 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  CC )
983ad2ant1 987 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  e.  CC )
109adantr 274 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  e.  CC )
11 simpl2 970 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  =/=  0
)
12 0z 9033 . . . . . . . . . . . . 13  |-  0  e.  ZZ
13 zapne 9093 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  ->  ( M #  0  <->  M  =/=  0 ) )
1412, 13mpan2 421 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  ( M #  0  <->  M  =/=  0
) )
15143ad2ant1 987 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M #  0  <->  M  =/=  0
) )
1615adantr 274 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( M #  0  <-> 
M  =/=  0 ) )
1711, 16mpbird 166 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M #  0 )
185, 7, 10, 17divmulap3d 8553 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  N  =  (
k  x.  M ) ) )
19 eqcom 2119 . . . . . . . 8  |-  ( N  =  ( k  x.  M )  <->  ( k  x.  M )  =  N )
2018, 19syl6bb 195 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  ( k  x.  M )  =  N ) )
2120biimprd 157 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( k  x.  M )  =  N  ->  ( N  /  M )  =  k ) )
2221impr 376 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  =  k )
23 simprl 505 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
k  e.  ZZ )
2422, 23eqeltrd 2194 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  e.  ZZ )
2524rexlimdvaa 2527 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
26 simpr 109 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( N  /  M )  e.  ZZ )
27 simp2 967 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  =/=  0 )
2827, 15mpbird 166 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M #  0 )
294, 9, 28divcanap1d 8519 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  x.  M )  =  N )
3029adantr 274 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( ( N  /  M )  x.  M )  =  N )
31 oveq1 5749 . . . . . . 7  |-  ( k  =  ( N  /  M )  ->  (
k  x.  M )  =  ( ( N  /  M )  x.  M ) )
3231eqeq1d 2126 . . . . . 6  |-  ( k  =  ( N  /  M )  ->  (
( k  x.  M
)  =  N  <->  ( ( N  /  M )  x.  M )  =  N ) )
3332rspcev 2763 . . . . 5  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( ( N  /  M )  x.  M
)  =  N )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
3426, 30, 33syl2anc 408 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
3534ex 114 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
3625, 35impbid 128 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  <->  ( N  /  M )  e.  ZZ ) )
372, 36bitrd 187 1  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285   E.wrex 2394   class class class wbr 3899  (class class class)co 5742   CCcc 7586   0cc0 7588    x. cmul 7593   # cap 8311    / cdiv 8400   ZZcz 9022    || cdvds 11420
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-n0 8946  df-z 9023  df-dvds 11421
This theorem is referenced by:  dvdsval3  11424  nndivdvds  11426  divconjdvds  11474  zeo3  11492  evend2  11513  oddp1d2  11514  fldivndvdslt  11559  divgcdz  11587  dvdsgcdidd  11609  mulgcd  11631  sqgcd  11644  lcmgcdlem  11685  mulgcddvds  11702  qredeu  11705  prmind2  11728  divgcdodd  11748  divnumden  11801  hashdvds  11824  hashgcdlem  11830
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