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Theorem dvdsval2 12176
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 12175 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
213adant2 1019 . 2 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
3 zcn 9397 . . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
433ad2ant3 1023 . . . . . . . . . 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 9397 . . . . . . . . . 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 9397 . . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
983ad2ant1 1021 . . . . . . . . . 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 1004 . . . . . . . . . 10 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M =/= 0 )
12 0z 9403 . . . . . . . . . . . . 13 |- 0 e. ZZ
13 zapne 9467 . . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
1412, 13mpan2 425 . . . . . . . . . . . 12 |- ( M e. ZZ -> ( M # 0 <-> M =/= 0 ) )
15143ad2ant1 1021 . . . . . . . . . . 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 8918 . . . . . . . 8 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> N = ( k x. M ) ) )
19 eqcom 2208 . . . . . . . 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 2283 . . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) e. ZZ )
2524rexlimdvaa 2625 . . 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 1001 . . . . . . . 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 8884 . . . . . 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 5964 . . . . . . 7 |- ( k = ( N / M ) -> ( k x. M ) = ( ( N / M ) x. M ) )
3231eqeq1d 2215 . . . . . 6 |- ( k = ( N / M ) -> ( ( k x. M ) = N <-> ( ( N / M ) x. M ) = N ) )
3332rspcev 2881 . . . . 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 981   = wceq 1373   e. wcel 2177   =/= wne 2377   E.wrex 2486   class class class wbr 4051  (class class class)co 5957   CCcc 7943   0cc0 7945   x. cmul 7950   # cap 8674   / cdiv 8765   ZZcz 9392   || cdvds 12173
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-id 4348  df-po 4351  df-iso 4352  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-n0 9316  df-z 9393  df-dvds 12174
This theorem is referenced by:  dvdsval3  12177  nndivdvds  12182  fsumdvds  12228  divconjdvds  12235  3dvds  12250  zeo3  12254  evend2  12275  oddp1d2  12276  fldivndvdslt  12323  bitsmod  12342  divgcdz  12367  dvdsgcdidd  12390  mulgcd  12412  sqgcd  12425  lcmgcdlem  12474  mulgcddvds  12491  qredeu  12494  prmind2  12517  isprm5lem  12538  divgcdodd  12540  divnumden  12593  hashdvds  12618  hashgcdlem  12635  pythagtriplem19  12680  pcprendvds2  12689  pcpremul  12691  pc2dvds  12728  pcz  12730  dvdsprmpweqle  12735  pcadd  12738  pcmptdvds  12743  fldivp1  12746  pockthlem  12754  4sqlem8  12783  4sqlem9  12784  4sqlem12  12800  4sqlem14  12802  znidomb  14495  lgseisenlem1  15622  lgsquad2lem1  15633  lgsquad3  15636  m1lgs  15637  2sqlem3  15669  2sqlem8  15675
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