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Theorem dvdsval2 12020
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 12019 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
213adant2 1018 . 2 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
3 zcn 9359 . . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
433ad2ant3 1022 . . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> N e. CC )
54adantr 276 . . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> N e. CC )
6 zcn 9359 . . . . . . . . . 10 |- ( k e. ZZ -> k e. CC )
76adantl 277 . . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> k e. CC )
8 zcn 9359 . . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
983ad2ant1 1020 . . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> M e. CC )
109adantr 276 . . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M e. CC )
11 simpl2 1003 . . . . . . . . . 10 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M =/= 0 )
12 0z 9365 . . . . . . . . . . . . 13 |- 0 e. ZZ
13 zapne 9429 . . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
1412, 13mpan2 425 . . . . . . . . . . . 12 |- ( M e. ZZ -> ( M # 0 <-> M =/= 0 ) )
15143ad2ant1 1020 . . . . . . . . . . 11 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
1615adantr 276 . . . . . . . . . 10 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
1711, 16mpbird 167 . . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M # 0 )
185, 7, 10, 17divmulap3d 8880 . . . . . . . 8 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> N = ( k x. M ) ) )
19 eqcom 2206 . . . . . . . 8 |- ( N = ( k x. M ) <-> ( k x. M ) = N )
2018, 19bitrdi 196 . . . . . . 7 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> ( k x. M ) = N ) )
2120biimprd 158 . . . . . 6 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( k x. M ) = N -> ( N / M ) = k ) )
2221impr 379 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) = k )
23 simprl 529 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> k e. ZZ )
2422, 23eqeltrd 2281 . . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) e. ZZ )
2524rexlimdvaa 2623 . . 3 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( E. k e. ZZ ( k x. M ) = N -> ( N / M ) e. ZZ ) )
26 simpr 110 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> ( N / M ) e. ZZ )
27 simp2 1000 . . . . . . . 8 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> M =/= 0 )
2827, 15mpbird 167 . . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> M # 0 )
294, 9, 28divcanap1d 8846 . . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( ( N / M ) x. M ) = N )
3029adantr 276 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> ( ( N / M ) x. M ) = N )
31 oveq1 5941 . . . . . . 7 |- ( k = ( N / M ) -> ( k x. M ) = ( ( N / M ) x. M ) )
3231eqeq1d 2213 . . . . . 6 |- ( k = ( N / M ) -> ( ( k x. M ) = N <-> ( ( N / M ) x. M ) = N ) )
3332rspcev 2876 . . . . 5 |- ( ( ( N / M ) e. ZZ /\ ( ( N / M ) x. M ) = N ) -> E. k e. ZZ ( k x. M ) = N )
3426, 30, 33syl2anc 411 . . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> E. k e. ZZ ( k x. M ) = N )
3534ex 115 . . 3 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( ( N / M ) e. ZZ -> E. k e. ZZ ( k x. M ) = N ) )
3625, 35impbid 129 . 2 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( E. k e. ZZ ( k x. M ) = N <-> ( N / M ) e. ZZ ) )
372, 36bitrd 188 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 104   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   =/= wne 2375   E.wrex 2484   class class class wbr 4043  (class class class)co 5934   CCcc 7905   0cc0 7907   x. cmul 7912   # cap 8636   / cdiv 8727   ZZcz 9354   || cdvds 12017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-id 4338  df-po 4341  df-iso 4342  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-iota 5229  df-fun 5270  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-n0 9278  df-z 9355  df-dvds 12018
This theorem is referenced by:  dvdsval3  12021  nndivdvds  12026  fsumdvds  12072  divconjdvds  12079  3dvds  12094  zeo3  12098  evend2  12119  oddp1d2  12120  fldivndvdslt  12167  bitsmod  12186  divgcdz  12211  dvdsgcdidd  12234  mulgcd  12256  sqgcd  12269  lcmgcdlem  12318  mulgcddvds  12335  qredeu  12338  prmind2  12361  isprm5lem  12382  divgcdodd  12384  divnumden  12437  hashdvds  12462  hashgcdlem  12479  pythagtriplem19  12524  pcprendvds2  12533  pcpremul  12535  pc2dvds  12572  pcz  12574  dvdsprmpweqle  12579  pcadd  12582  pcmptdvds  12587  fldivp1  12590  pockthlem  12598  4sqlem8  12627  4sqlem9  12628  4sqlem12  12644  4sqlem14  12646  znidomb  14338  lgseisenlem1  15465  lgsquad2lem1  15476  lgsquad3  15479  m1lgs  15480  2sqlem3  15512  2sqlem8  15518
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