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Theorem dvdsval2 12582
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 12581 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
213adant2 1047 . 2 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
3 zcn 9660 . . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
433ad2ant3 1051 . . . . . . . . . 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 9660 . . . . . . . . . 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 9660 . . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
983ad2ant1 1049 . . . . . . . . . 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 1032 . . . . . . . . . 10 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M =/= 0 )
12 0z 9666 . . . . . . . . . . . . 13 |- 0 e. ZZ
13 zapne 9730 . . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
1412, 13mpan2 429 . . . . . . . . . . . 12 |- ( M e. ZZ -> ( M # 0 <-> M =/= 0 ) )
15143ad2ant1 1049 . . . . . . . . . . 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 9163 . . . . . . . 8 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> N = ( k x. M ) ) )
19 eqcom 2240 . . . . . . . 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 535 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> k e. ZZ )
2422, 23eqeltrd 2315 . . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) e. ZZ )
2524rexlimdvaa 2669 . . 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 1029 . . . . . . . 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 9129 . . . . . 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 6095 . . . . . . 7 |- ( k = ( N / M ) -> ( k x. M ) = ( ( N / M ) x. M ) )
3231eqeq1d 2247 . . . . . 6 |- ( k = ( N / M ) -> ( ( k x. M ) = N <-> ( ( N / M ) x. M ) = N ) )
3332rspcev 2929 . . . . 5 |- ( ( ( N / M ) e. ZZ /\ ( ( N / M ) x. M ) = N ) -> E. k e. ZZ ( k x. M ) = N )
3426, 30, 33syl2anc 415 . . . 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   class class class wbr 4131  (class class class)co 6088   CCcc 8183   0cc0 8185   x. cmul 8190   # cap 8917   / cdiv 9010   ZZcz 9655   || cdvds 12579
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-dvds 12580
This theorem is used by:  dvdsval3  12583  nndivdvds  12588  fsumdvds  12634  divconjdvds  12641  3dvds  12656  zeo3  12660  evend2  12681  oddp1d2  12682  fldivndvdslt  12729  bitsmod  12748  divgcdz  12773  dvdsgcdidd  12796  mulgcd  12818  sqgcd  12831  lcmgcdlem  12880  mulgcddvds  12897  qredeu  12900  prmind2  12923  isprm5lem  12945  divgcdodd  12947  divnumden  13001  hashdvds  13028  hashgcdlem  13045  pythagtriplem19  13090  pcprendvds2  13099  pcpremul  13101  pc2dvds  13138  pcz  13140  dvdsprmpweqle  13145  pcadd  13148  pcmptdvds  13153  fldivp1  13156  pockthlem  13164  4sqlem8  13193  4sqlem9  13194  4sqlem12  13210  4sqlem14  13212  znidomb  15102  efchtqdvds  16265  bposlem6  16316  lgseisenlem1  16394  lgsquad2lem1  16405  lgsquad3  16408  m1lgs  16409  2sqlem3  16441  2sqlem8  16447
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