Theorem dvdsval2 10198

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.

Step	Hyp	Ref	Expression
1		divides 10197	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
2	1	3adant2 957	. 2 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> E. k e. ZZ ( k x. M ) = N ) )
3		zcn 8356	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
4	3	3ad2ant3 961	. . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> N e. CC )
5	4	adantr 270	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> N e. CC )
6		zcn 8356	. . . . . . . . . 10 |- ( k e. ZZ -> k e. CC )
7	6	adantl 271	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> k e. CC )
8		zcn 8356	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
9	8	3ad2ant1 959	. . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> M e. CC )
10	9	adantr 270	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M e. CC )
11		simpl2 942	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M =/= 0 )
12		0z 8362	. . . . . . . . . . . . 13 |- 0 e. ZZ
13		zapne 8422	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
14	12, 13	mpan2 415	. . . . . . . . . . . 12 |- ( M e. ZZ -> ( M # 0 <-> M =/= 0 ) )
15	14	3ad2ant1 959	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
16	15	adantr 270	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
17	11, 16	mpbird 165	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> M # 0 )
18	5, 7, 10, 17	divmulap3d 7911	. . . . . . . 8 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> N = ( k x. M ) ) )
19		eqcom 2083	. . . . . . . 8 |- ( N = ( k x. M ) <-> ( k x. M ) = N )
20	18, 19	syl6bb 194	. . . . . . 7 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( N / M ) = k <-> ( k x. M ) = N ) )
21	20	biimprd 156	. . . . . 6 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ k e. ZZ ) -> ( ( k x. M ) = N -> ( N / M ) = k ) )
22	21	impr 371	. . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) = k )
23		simprl 497	. . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> k e. ZZ )
24	22, 23	eqeltrd 2155	. . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( k e. ZZ /\ ( k x. M ) = N ) ) -> ( N / M ) e. ZZ )
25	24	rexlimdvaa 2478	. . 3 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( E. k e. ZZ ( k x. M ) = N -> ( N / M ) e. ZZ ) )
26		simpr 108	. . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> ( N / M ) e. ZZ )
27		simp2 939	. . . . . . . 8 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> M =/= 0 )
28	27, 15	mpbird 165	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> M # 0 )
29	4, 9, 28	divcanap1d 7878	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( ( N / M ) x. M ) = N )
30	29	adantr 270	. . . . 5 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> ( ( N / M ) x. M ) = N )
31		oveq1 5539	. . . . . . 7 |- ( k = ( N / M ) -> ( k x. M ) = ( ( N / M ) x. M ) )
32	31	eqeq1d 2089	. . . . . 6 |- ( k = ( N / M ) -> ( ( k x. M ) = N <-> ( ( N / M ) x. M ) = N ) )
33	32	rspcev 2701	. . . . 5 |- ( ( ( N / M ) e. ZZ /\ ( ( N / M ) x. M ) = N ) -> E. k e. ZZ ( k x. M ) = N )
34	26, 30, 33	syl2anc 403	. . . 4 |- ( ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) /\ ( N / M ) e. ZZ ) -> E. k e. ZZ ( k x. M ) = N )
35	34	ex 113	. . 3 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( ( N / M ) e. ZZ -> E. k e. ZZ ( k x. M ) = N ) )
36	25, 35	impbid 127	. 2 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( E. k e. ZZ ( k x. M ) = N <-> ( N / M ) e. ZZ ) )
37	2, 36	bitrd 186	1 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> ( N / M ) e. ZZ ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   =/= wne 2245   E.wrex 2349   class class class wbr 3785  (class class class)co 5532   CCcc 6979   0cc0 6981   x. cmul 6986   # cap 7681   / cdiv 7760   ZZcz 8351   || cdvds 10195

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-dvds 10196

This theorem is referenced by:  dvdsval3  10199  nndivdvds  10201  divconjdvds  10249  zeo3  10267  evend2  10289  oddp1d2  10290  fldivndvdslt  10335  divgcdz  10363  mulgcd  10405  sqgcd  10418  lcmgcdlem  10459  mulgcddvds  10476  qredeu  10479  prmind2  10502  divgcdodd  10522