Theorem negdvdsb 10211

Description: An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
negdvdsb	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> -u M || N ) )

Proof of Theorem negdvdsb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
2		znegcl 8382	. . . 4 |- ( M e. ZZ -> -u M e. ZZ )
3	2	anim1i 333	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M e. ZZ /\ N e. ZZ ) )
4		znegcl 8382	. . . 4 |- ( x e. ZZ -> -u x e. ZZ )
5	4	adantl 271	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> -u x e. ZZ )
6		zcn 8356	. . . . . . 7 |- ( x e. ZZ -> x e. CC )
7		zcn 8356	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		mul2neg 7502	. . . . . . 7 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. -u M ) = ( x x. M ) )
9	6, 7, 8	syl2anr 284	. . . . . 6 |- ( ( M e. ZZ /\ x e. ZZ ) -> ( -u x x. -u M ) = ( x x. M ) )
10	9	adantlr 460	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( -u x x. -u M ) = ( x x. M ) )
11	10	eqeq1d 2089	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( -u x x. -u M ) = N <-> ( x x. M ) = N ) )
12	11	biimprd 156	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( -u x x. -u M ) = N ) )
13	1, 3, 5, 12	dvds1lem 10206	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> -u M || N ) )
14		mulneg12 7501	. . . . . . 7 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. M ) = ( x x. -u M ) )
15	6, 7, 14	syl2anr 284	. . . . . 6 |- ( ( M e. ZZ /\ x e. ZZ ) -> ( -u x x. M ) = ( x x. -u M ) )
16	15	adantlr 460	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( -u x x. M ) = ( x x. -u M ) )
17	16	eqeq1d 2089	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( -u x x. M ) = N <-> ( x x. -u M ) = N ) )
18	17	biimprd 156	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. -u M ) = N -> ( -u x x. M ) = N ) )
19	3, 1, 5, 18	dvds1lem 10206	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M || N -> M || N ) )
20	13, 19	impbid 127	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> -u M || N ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   x. cmul 6986   -ucneg 7280   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-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

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-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-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-inn 8040  df-z 8352  df-dvds 10196

This theorem is referenced by:  absdvdsb  10213  zdvdsdc  10216  bezoutlemzz  10391  lcmneg  10456