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Theorem dvdsval2 12534
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 12533 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
213adant2 974 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  ( k  x.  M )  =  N ) )
3 zcn 10029 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  N  e.  CC )
433ad2ant3 978 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  N  e.  CC )
54adantr 451 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  N  e.  CC )
6 zcn 10029 . . . . . . . . . . 11  |-  ( k  e.  ZZ  ->  k  e.  CC )
76adantl 452 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  k  e.  CC )
8 zcn 10029 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  CC )
983ad2ant1 976 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  e.  CC )
109adantr 451 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  e.  CC )
11 simpl2 959 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  =/=  0
)
125, 7, 10, 11divmul3d 9570 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  N  =  (
k  x.  M ) ) )
13 eqcom 2285 . . . . . . . . 9  |-  ( N  =  ( k  x.  M )  <->  ( k  x.  M )  =  N )
1412, 13syl6bb 252 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  ( k  x.  M )  =  N ) )
1514biimprd 214 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( k  x.  M )  =  N  ->  ( N  /  M )  =  k ) )
1615impr 602 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  =  k )
17 simprl 732 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
k  e.  ZZ )
1816, 17eqeltrd 2357 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  e.  ZZ )
1918exp32 588 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
k  e.  ZZ  ->  ( ( k  x.  M
)  =  N  -> 
( N  /  M
)  e.  ZZ ) ) )
2019rexlimdv 2666 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
21 simpr 447 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( N  /  M )  e.  ZZ )
22 simp2 956 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  =/=  0 )
234, 9, 22divcan1d 9537 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  x.  M )  =  N )
2423adantr 451 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( ( N  /  M )  x.  M )  =  N )
25 oveq1 5865 . . . . . . 7  |-  ( k  =  ( N  /  M )  ->  (
k  x.  M )  =  ( ( N  /  M )  x.  M ) )
2625eqeq1d 2291 . . . . . 6  |-  ( k  =  ( N  /  M )  ->  (
( k  x.  M
)  =  N  <->  ( ( N  /  M )  x.  M )  =  N ) )
2726rspcev 2884 . . . . 5  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( ( N  /  M )  x.  M
)  =  N )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
2821, 24, 27syl2anc 642 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
2928ex 423 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
3020, 29impbid 183 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  <->  ( N  /  M )  e.  ZZ ) )
312, 30bitrd 244 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023  (class class class)co 5858   CCcc 8735   0cc0 8737    x. cmul 8742    / cdiv 9423   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  dvdsval3  12535  nndivdvds  12537  fsumdvds  12572  3dvds  12591  bitsmod  12627  sadaddlem  12657  bitsuz  12665  mulgcd  12725  sqgcd  12737  prmind2  12769  mulgcddvds  12783  qredeu  12786  isprm5  12791  divgcdodd  12798  divnumden  12819  hashdvds  12843  oddprm  12868  pythagtriplem11  12878  pythagtriplem13  12880  pythagtriplem19  12886  pcprendvds2  12894  pcpremul  12896  pc2dvds  12931  pcz  12933  pcadd  12937  pcmptdvds  12942  fldivp1  12945  pockthlem  12952  prmreclem1  12963  prmreclem3  12965  4sqlem8  12992  4sqlem9  12993  4sqlem12  13003  4sqlem14  13005  sylow1lem1  14909  sylow3lem4  14941  odadd1  15140  odadd2  15141  pgpfac1lem3  15312  prmirredlem  16446  znidomb  16515  root1eq1  20095  atantayl2  20234  efchtdvds  20397  dvdsdivcl  20421  muinv  20433  chtub  20451  bposlem6  20528  lgseisenlem1  20588  lgsquad2lem1  20597  lgsquad3  20600  m1lgs  20601  2sqlem3  20605  2sqlem8  20611  divides2OLD  26237  nn0prpwlem  26238  congrep  27060  jm2.22  27088  jm2.23  27089  hashgcdlem  27516  proot1ex  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-z 10025  df-dvds 12532
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