Theorem zdiv 9734

Description: Two ways to express " M divides N. (Contributed by NM, 3-Oct-2008.)

Assertion

Ref	Expression
zdiv	|- ( ( M e. NN /\ N e. ZZ ) -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) )

Distinct variable groups:   k, M   k, N

Proof of Theorem zdiv

Step	Hyp	Ref	Expression
1		nnap0 9333	. . 3 |- ( M e. NN -> M # 0 )
2	1	adantr 276	. 2 |- ( ( M e. NN /\ N e. ZZ ) -> M # 0 )
3		nncn 9312	. . 3 |- ( M e. NN -> M e. CC )
4		zcn 9649	. . 3 |- ( N e. ZZ -> N e. CC )
5		zcn 9649	. . . . . . . . . . 11 |- ( k e. ZZ -> k e. CC )
6		divcanap3 9028	. . . . . . . . . . . . 13 |- ( ( k e. CC /\ M e. CC /\ M # 0 ) -> ( ( M x. k ) / M ) = k )
7	6	3coml 1241	. . . . . . . . . . . 12 |- ( ( M e. CC /\ M # 0 /\ k e. CC ) -> ( ( M x. k ) / M ) = k )
8	7	3expa 1234	. . . . . . . . . . 11 |- ( ( ( M e. CC /\ M # 0 ) /\ k e. CC ) -> ( ( M x. k ) / M ) = k )
9	5, 8	sylan2 286	. . . . . . . . . 10 |- ( ( ( M e. CC /\ M # 0 ) /\ k e. ZZ ) -> ( ( M x. k ) / M ) = k )
10	9	3adantl2 1185	. . . . . . . . 9 |- ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) -> ( ( M x. k ) / M ) = k )
11		oveq1 6092	. . . . . . . . 9 |- ( ( M x. k ) = N -> ( ( M x. k ) / M ) = ( N / M ) )
12	10, 11	sylan9req 2292	. . . . . . . 8 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k = ( N / M ) )
13		simplr 533	. . . . . . . 8 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k e. ZZ )
14	12, 13	eqeltrrd 2316	. . . . . . 7 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> ( N / M ) e. ZZ )
15	14	exp31 364	. . . . . 6 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( k e. ZZ -> ( ( M x. k ) = N -> ( N / M ) e. ZZ ) ) )
16	15	rexlimdv 2667	. . . . 5 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( E. k e. ZZ ( M x. k ) = N -> ( N / M ) e. ZZ ) )
17		divcanap2 9010	. . . . . . 7 |- ( ( N e. CC /\ M e. CC /\ M # 0 ) -> ( M x. ( N / M ) ) = N )
18	17	3com12 1238	. . . . . 6 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( M x. ( N / M ) ) = N )
19		oveq2 6093	. . . . . . . . 9 |- ( k = ( N / M ) -> ( M x. k ) = ( M x. ( N / M ) ) )
20	19	eqeq1d 2247	. . . . . . . 8 |- ( k = ( N / M ) -> ( ( M x. k ) = N <-> ( M x. ( N / M ) ) = N ) )
21	20	rspcev 2929	. . . . . . 7 |- ( ( ( N / M ) e. ZZ /\ ( M x. ( N / M ) ) = N ) -> E. k e. ZZ ( M x. k ) = N )
22	21	expcom 116	. . . . . 6 |- ( ( M x. ( N / M ) ) = N -> ( ( N / M ) e. ZZ -> E. k e. ZZ ( M x. k ) = N ) )
23	18, 22	syl 14	. . . . 5 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( ( N / M ) e. ZZ -> E. k e. ZZ ( M x. k ) = N ) )
24	16, 23	impbid 129	. . . 4 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) )
25	24	3expia 1236	. . 3 |- ( ( M e. CC /\ N e. CC ) -> ( M # 0 -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) ) )
26	3, 4, 25	syl2an 289	. 2 |- ( ( M e. NN /\ N e. ZZ ) -> ( M # 0 -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) ) )
27	2, 26	mpd 13	1 |- ( ( M e. NN /\ N e. ZZ ) -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130  (class class class)co 6085   CCcc 8177   0cc0 8179   x. cmul 8184   # cap 8909   / cdiv 9002   NNcn 9304   ZZcz 9644

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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

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 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-z 9645

This theorem is used by:  addmodlteq  10835