Theorem zdiv 9461

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 9065	. . 3 |- ( M e. NN -> M # 0 )
2	1	adantr 276	. 2 |- ( ( M e. NN /\ N e. ZZ ) -> M # 0 )
3		nncn 9044	. . 3 |- ( M e. NN -> M e. CC )
4		zcn 9377	. . 3 |- ( N e. ZZ -> N e. CC )
5		zcn 9377	. . . . . . . . . . 11 |- ( k e. ZZ -> k e. CC )
6		divcanap3 8771	. . . . . . . . . . . . 13 |- ( ( k e. CC /\ M e. CC /\ M # 0 ) -> ( ( M x. k ) / M ) = k )
7	6	3coml 1213	. . . . . . . . . . . 12 |- ( ( M e. CC /\ M # 0 /\ k e. CC ) -> ( ( M x. k ) / M ) = k )
8	7	3expa 1206	. . . . . . . . . . 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 1157	. . . . . . . . 9 |- ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) -> ( ( M x. k ) / M ) = k )
11		oveq1 5951	. . . . . . . . 9 |- ( ( M x. k ) = N -> ( ( M x. k ) / M ) = ( N / M ) )
12	10, 11	sylan9req 2259	. . . . . . . 8 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k = ( N / M ) )
13		simplr 528	. . . . . . . 8 |- ( ( ( ( M e. CC /\ N e. CC /\ M # 0 ) /\ k e. ZZ ) /\ ( M x. k ) = N ) -> k e. ZZ )
14	12, 13	eqeltrrd 2283	. . . . . . 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 2622	. . . . 5 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( E. k e. ZZ ( M x. k ) = N -> ( N / M ) e. ZZ ) )
17		divcanap2 8753	. . . . . . 7 |- ( ( N e. CC /\ M e. CC /\ M # 0 ) -> ( M x. ( N / M ) ) = N )
18	17	3com12 1210	. . . . . 6 |- ( ( M e. CC /\ N e. CC /\ M # 0 ) -> ( M x. ( N / M ) ) = N )
19		oveq2 5952	. . . . . . . . 9 |- ( k = ( N / M ) -> ( M x. k ) = ( M x. ( N / M ) ) )
20	19	eqeq1d 2214	. . . . . . . 8 |- ( k = ( N / M ) -> ( ( M x. k ) = N <-> ( M x. ( N / M ) ) = N ) )
21	20	rspcev 2877	. . . . . . 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 1208	. . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4044  (class class class)co 5944   CCcc 7923   0cc0 7925   x. cmul 7930   # cap 8654   / cdiv 8745   NNcn 9036   ZZcz 9372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-z 9373

This theorem is referenced by:  addmodlteq  10543