Theorem zdiv 9258

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 8868	. . 3 |- ( M e. NN -> M # 0 )
2	1	adantr 274	. 2 |- ( ( M e. NN /\ N e. ZZ ) -> M # 0 )
3		nncn 8847	. . 3 |- ( M e. NN -> M e. CC )
4		zcn 9178	. . 3 |- ( N e. ZZ -> N e. CC )
5		zcn 9178	. . . . . . . . . . 11 |- ( k e. ZZ -> k e. CC )
6		divcanap3 8576	. . . . . . . . . . . . 13 |- ( ( k e. CC /\ M e. CC /\ M # 0 ) -> ( ( M x. k ) / M ) = k )
7	6	3coml 1192	. . . . . . . . . . . 12 |- ( ( M e. CC /\ M # 0 /\ k e. CC ) -> ( ( M x. k ) / M ) = k )
8	7	3expa 1185	. . . . . . . . . . 11 |- ( ( ( M e. CC /\ M # 0 ) /\ k e. CC ) -> ( ( M x. k ) / M ) = k )
9	5, 8	sylan2 284	. . . . . . . . . 10 |- ( ( ( M e. CC /\ M # 0 ) /\ k e. ZZ ) -> ( ( M x. k ) / M ) = k )
10	9	3adantl2 1139	. . . . . . . . 9 |- ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) -> ( ( M x. k ) / M ) = k )
11		oveq1 5834	. . . . . . . . 9 |- ( ( M x. k ) = N -> ( ( M x. k ) / M ) = ( N / M ) )
12	10, 11	sylan9req 2211	. . . . . . . 8 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k = ( N / M ) )
13		simplr 520	. . . . . . . 8 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k e. ZZ )
14	12, 13	eqeltrrd 2235	. . . . . . 7 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> ( N / M ) e. ZZ )
15	14	exp31 362	. . . . . 6 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( k e. ZZ -> ( ( M x. k ) = N -> ( N / M ) e. ZZ ) ) )
16	15	rexlimdv 2573	. . . . 5 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( E. k e. ZZ ( M x. k ) = N -> ( N / M ) e. ZZ ) )
17		divcanap2 8558	. . . . . . 7 |- ( ( N e. CC /\ M e. CC /\ M # 0 ) -> ( M x. ( N / M ) ) = N )
18	17	3com12 1189	. . . . . 6 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( M x. ( N / M ) ) = N )
19		oveq2 5835	. . . . . . . . 9 |- ( k = ( N / M ) -> ( M x. k ) = ( M x. ( N / M ) ) )
20	19	eqeq1d 2166	. . . . . . . 8 |- ( k = ( N / M ) -> ( ( M x. k ) = N <-> ( M x. ( N / M ) ) = N ) )
21	20	rspcev 2816	. . . . . . 7 |- ( ( ( N / M ) e. ZZ /\ ( M x. ( N / M ) ) = N ) -> E. k e. ZZ ( M x. k ) = N )
22	21	expcom 115	. . . . . 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 128	. . . 4 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) )
25	24	3expia 1187	. . 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 287	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1335   e. wcel 2128   E.wrex 2436   class class class wbr 3967  (class class class)co 5827   CCcc 7733   0cc0 7735   x. cmul 7740   # cap 8461   / cdiv 8550   NNcn 8839   ZZcz 9173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4029  df-id 4256  df-po 4259  df-iso 4260  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-iota 5138  df-fun 5175  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-z 9174

This theorem is referenced by:  addmodlteq  10307