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Theorem dvdsval2 12782
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 12781 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
213adant2 976 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  ( k  x.  M )  =  N ) )
3 zcn 10219 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
433ad2ant3 980 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  N  e.  CC )
54adantr 452 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  N  e.  CC )
6 zcn 10219 . . . . . . . . . 10  |-  ( k  e.  ZZ  ->  k  e.  CC )
76adantl 453 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  k  e.  CC )
8 zcn 10219 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  CC )
983ad2ant1 978 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  e.  CC )
109adantr 452 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  e.  CC )
11 simpl2 961 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  =/=  0
)
125, 7, 10, 11divmul3d 9756 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  N  =  (
k  x.  M ) ) )
13 eqcom 2389 . . . . . . . 8  |-  ( N  =  ( k  x.  M )  <->  ( k  x.  M )  =  N )
1412, 13syl6bb 253 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  ( k  x.  M )  =  N ) )
1514biimprd 215 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( k  x.  M )  =  N  ->  ( N  /  M )  =  k ) )
1615impr 603 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  =  k )
17 simprl 733 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
k  e.  ZZ )
1816, 17eqeltrd 2461 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  e.  ZZ )
1918rexlimdvaa 2774 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
20 simpr 448 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( N  /  M )  e.  ZZ )
21 simp2 958 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  =/=  0 )
224, 9, 21divcan1d 9723 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  x.  M )  =  N )
2322adantr 452 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( ( N  /  M )  x.  M )  =  N )
24 oveq1 6027 . . . . . . 7  |-  ( k  =  ( N  /  M )  ->  (
k  x.  M )  =  ( ( N  /  M )  x.  M ) )
2524eqeq1d 2395 . . . . . 6  |-  ( k  =  ( N  /  M )  ->  (
( k  x.  M
)  =  N  <->  ( ( N  /  M )  x.  M )  =  N ) )
2625rspcev 2995 . . . . 5  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( ( N  /  M )  x.  M
)  =  N )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
2720, 23, 26syl2anc 643 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
2827ex 424 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
2919, 28impbid 184 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  <->  ( N  /  M )  e.  ZZ ) )
302, 29bitrd 245 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650   class class class wbr 4153  (class class class)co 6020   CCcc 8921   0cc0 8923    x. cmul 8928    / cdiv 9609   ZZcz 10214    || cdivides 12779
This theorem is referenced by:  dvdsval3  12783  nndivdvds  12785  fsumdvds  12820  3dvds  12839  bitsmod  12875  sadaddlem  12905  bitsuz  12913  mulgcd  12973  sqgcd  12985  prmind2  13017  mulgcddvds  13031  qredeu  13034  isprm5  13039  divgcdodd  13046  divnumden  13067  hashdvds  13091  oddprm  13116  pythagtriplem11  13126  pythagtriplem13  13128  pythagtriplem19  13134  pcprendvds2  13142  pcpremul  13144  pc2dvds  13179  pcz  13181  pcadd  13185  pcmptdvds  13190  fldivp1  13193  pockthlem  13200  prmreclem1  13211  prmreclem3  13213  4sqlem8  13240  4sqlem9  13241  4sqlem12  13251  4sqlem14  13253  sylow1lem1  15159  sylow3lem4  15191  odadd1  15390  odadd2  15391  pgpfac1lem3  15562  prmirredlem  16696  znidomb  16765  root1eq1  20506  atantayl2  20645  efchtdvds  20809  dvdsdivcl  20833  muinv  20845  chtub  20863  bposlem6  20940  lgseisenlem1  21000  lgsquad2lem1  21009  lgsquad3  21012  m1lgs  21013  2sqlem3  21017  2sqlem8  21023  qqhval2lem  24164  nn0prpwlem  26016  congrep  26729  jm2.22  26757  jm2.23  26758  hashgcdlem  27185  proot1ex  27189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-z 10215  df-dvds 12780
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