Theorem zdiv 9300

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 8907	. . 3 |- ( M e. NN -> M # 0 )
2	1	adantr 274	. 2 |- ( ( M e. NN /\ N e. ZZ ) -> M # 0 )
3		nncn 8886	. . 3 |- ( M e. NN -> M e. CC )
4		zcn 9217	. . 3 |- ( N e. ZZ -> N e. CC )
5		zcn 9217	. . . . . . . . . . 11 |- ( k e. ZZ -> k e. CC )
6		divcanap3 8615	. . . . . . . . . . . . 13 |- ( ( k e. CC /\ M e. CC /\ M # 0 ) -> ( ( M x. k ) / M ) = k )
7	6	3coml 1205	. . . . . . . . . . . 12 |- ( ( M e. CC /\ M # 0 /\ k e. CC ) -> ( ( M x. k ) / M ) = k )
8	7	3expa 1198	. . . . . . . . . . 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 1149	. . . . . . . . 9 |- ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) -> ( ( M x. k ) / M ) = k )
11		oveq1 5860	. . . . . . . . 9 |- ( ( M x. k ) = N -> ( ( M x. k ) / M ) = ( N / M ) )
12	10, 11	sylan9req 2224	. . . . . . . 8 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k = ( N / M ) )
13		simplr 525	. . . . . . . 8 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k e. ZZ )
14	12, 13	eqeltrrd 2248	. . . . . . 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 2586	. . . . 5 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( E. k e. ZZ ( M x. k ) = N -> ( N / M ) e. ZZ ) )
17		divcanap2 8597	. . . . . . 7 |- ( ( N e. CC /\ M e. CC /\ M # 0 ) -> ( M x. ( N / M ) ) = N )
18	17	3com12 1202	. . . . . 6 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( M x. ( N / M ) ) = N )
19		oveq2 5861	. . . . . . . . 9 |- ( k = ( N / M ) -> ( M x. k ) = ( M x. ( N / M ) ) )
20	19	eqeq1d 2179	. . . . . . . 8 |- ( k = ( N / M ) -> ( ( M x. k ) = N <-> ( M x. ( N / M ) ) = N ) )
21	20	rspcev 2834	. . . . . . 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 1200	. . 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 973   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   x. cmul 7779   # cap 8500   / cdiv 8589   NNcn 8878   ZZcz 9212

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-z 9213

This theorem is referenced by:  addmodlteq  10354