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Theorem dvdsval2 14986
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 14985 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
213adant2 1080 . 2 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
3 zcn 11382 . . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
433ad2ant3 1084 . . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> N e. CC )
54adantr 481 . . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> N e. CC )
6 zcn 11382 . . . . . . . . . 10 |- ( k e. ZZ -> k e. CC )
76adantl 482 . . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> k e. CC )
8 zcn 11382 . . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
983ad2ant1 1082 . . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> M e. CC )
109adantr 481 . . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M e. CC )
11 simpl2 1065 . . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M =/= 0 )
125, 7, 10, 11divmul3d 10835 . . . . . . . 8 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> N = ( k x. M ) ) )
13 eqcom 2629 . . . . . . . 8 |- ( N = ( k x. M ) <-> ( k x. M ) = N )
1412, 13syl6bb 276 . . . . . . 7 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> ( k x. M ) = N ) )
1514biimprd 238 . . . . . 6 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( k x. M ) = N -> ( N / M ) = k ) )
1615impr 649 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) = k )
17 simprl 794 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> k e. ZZ )
1816, 17eqeltrd 2701 . . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) e. ZZ )
1918rexlimdvaa 3032 . . 3 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( E. k e. ZZ ( k x. M ) = N -> ( N / M ) e. ZZ ) )
20 simpr 477 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> ( N / M ) e. ZZ )
21 simp2 1062 . . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> M =/= 0 )
224, 9, 21divcan1d 10802 . . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( ( N / M ) x. M ) = N )
2322adantr 481 . . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> ( ( N / M ) x. M ) = N )
24 oveq1 6657 . . . . . . 7 |- ( k = ( N / M ) -> ( k x. M ) = ( ( N / M ) x. M ) )
2524eqeq1d 2624 . . . . . 6 |- ( k = ( N / M ) -> ( ( k x. M ) = N <-> ( ( N / M ) x. M ) = N ) )
2625rspcev 3309 . . . . 5 |- ( ( ( N / M ) e. ZZ /\ ( ( N / M ) x. M ) = N ) -> E. k e. ZZ ( k x. M ) = N )
2720, 23, 26syl2anc 693 . . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> E. k e. ZZ ( k x. M ) = N )
2827ex 450 . . 3 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( ( N / M ) e. ZZ -> E. k e. ZZ ( k x. M ) = N ) )
2919, 28impbid 202 . 2 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( E. k e. ZZ ( k x. M ) = N <-> ( N / M ) e. ZZ ) )
302, 29bitrd 268 1 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> ( N / M ) e. ZZ ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   / cdiv 10684   ZZcz 11377   || cdvds 14983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-z 11378  df-dvds 14984
This theorem is referenced by:  dvdsval3  14987  nndivdvds  14989  fsumdvds  15030  divconjdvds  15037  3dvds  15052  3dvdsOLD  15053  evend2  15081  oddp1d2  15082  flodddiv4  15137  fldivndvdslt  15138  bitsmod  15158  sadaddlem  15188  bitsuz  15196  divgcdz  15233  mulgcd  15265  sqgcd  15278  lcmgcdlem  15319  mulgcddvds  15369  qredeu  15372  prmind2  15398  isprm5  15419  divgcdodd  15422  divnumden  15456  hashdvds  15480  hashgcdlem  15493  oddprm  15515  pythagtriplem11  15530  pythagtriplem13  15532  pythagtriplem19  15538  pcprendvds2  15546  pcpremul  15548  pc2dvds  15583  pcz  15585  dvdsprmpweqle  15590  pcadd  15593  pcmptdvds  15598  fldivp1  15601  pockthlem  15609  prmreclem1  15620  prmreclem3  15622  4sqlem8  15649  4sqlem9  15650  4sqlem12  15660  4sqlem14  15662  sylow1lem1  18013  sylow3lem4  18045  odadd1  18251  odadd2  18252  pgpfac1lem3  18476  prmirredlem  19841  znidomb  19910  root1eq1  24496  atantayl2  24665  efchtdvds  24885  muinv  24919  chtub  24937  bposlem6  25014  lgseisenlem1  25100  lgsquad2lem1  25109  lgsquad3  25112  m1lgs  25113  2sqlem3  25145  2sqlem8  25151  qqhval2lem  30025  nn0prpwlem  32317  knoppndvlem8  32510  congrep  37540  jm2.22  37562  jm2.23  37563  proot1ex  37779  nzss  38516  etransclem9  40460  etransclem38  40489  etransclem44  40495  etransclem45  40496  divgcdoddALTV  41593  0dig2nn0o  42407
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