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Theorem dvdsval2 11936
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 11935 . . 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 9325 . . . . . . . . . . 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 9325 . . . . . . . . . 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 9325 . . . . . . . . . . 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 9331 . . . . . . . . . . . . 13 |- 0 e. ZZ
13 zapne 9394 . . . . . . . . . . . . 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 8846 . . . . . . . 8 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> N = ( k x. M ) ) )
19 eqcom 2195 . . . . . . . 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 2270 . . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) e. ZZ )
2524rexlimdvaa 2612 . . 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 8812 . . . . . 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 5926 . . . . . . 7 |- ( k = ( N / M ) -> ( k x. M ) = ( ( N / M ) x. M ) )
3231eqeq1d 2202 . . . . . 6 |- ( k = ( N / M ) -> ( ( k x. M ) = N <-> ( ( N / M ) x. M ) = N ) )
3332rspcev 2865 . . . . 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 1364   e. wcel 2164   =/= wne 2364   E.wrex 2473   class class class wbr 4030  (class class class)co 5919   CCcc 7872   0cc0 7874   x. cmul 7879   # cap 8602   / cdiv 8693   ZZcz 9320   || cdvds 11933
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-dvds 11934
This theorem is referenced by:  dvdsval3  11937  nndivdvds  11942  divconjdvds  11994  zeo3  12012  evend2  12033  oddp1d2  12034  fldivndvdslt  12079  divgcdz  12111  dvdsgcdidd  12134  mulgcd  12156  sqgcd  12169  lcmgcdlem  12218  mulgcddvds  12235  qredeu  12238  prmind2  12261  isprm5lem  12282  divgcdodd  12284  divnumden  12337  hashdvds  12362  hashgcdlem  12379  pythagtriplem19  12423  pcprendvds2  12432  pcpremul  12434  pc2dvds  12471  pcz  12473  dvdsprmpweqle  12478  pcadd  12481  pcmptdvds  12486  fldivp1  12489  pockthlem  12497  4sqlem8  12526  4sqlem9  12527  4sqlem12  12543  4sqlem14  12545  znidomb  14157  lgseisenlem1  15227  lgsquad2lem1  15238  lgsquad3  15241  m1lgs  15242  2sqlem3  15274  2sqlem8  15280
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