Theorem negdvdsb 11090

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 8781	. . . 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 8781	. . . 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 8755	. . . . . . 7 |- ( x e. ZZ -> x e. CC )
7		zcn 8755	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		mul2neg 7876	. . . . . . 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 461	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( -u x x. -u M ) = ( x x. M ) )
11	10	eqeq1d 2096	. . . 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 11085	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> -u M || N ) )
14		mulneg12 7875	. . . . . . 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 461	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( -u x x. M ) = ( x x. -u M ) )
17	16	eqeq1d 2096	. . . 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 11085	. 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 1289   e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7348   x. cmul 7355   -ucneg 7654   ZZcz 8750   || cdvds 11074

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-z 8751  df-dvds 11075

This theorem is referenced by:  absdvdsb  11092  zdvdsdc  11095  bezoutlemzz  11269  lcmneg  11334