Theorem negdvdsb 12523

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 9629	. . . 4 |- ( M e. ZZ -> -u M e. ZZ )
3	2	anim1i 340	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M e. ZZ /\ N e. ZZ ) )
4		znegcl 9629	. . . 4 |- ( x e. ZZ -> -u x e. ZZ )
5	4	adantl 277	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> -u x e. ZZ )
6		zcn 9603	. . . . . . 7 |- ( x e. ZZ -> x e. CC )
7		zcn 9603	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		mul2neg 8690	. . . . . . 7 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. -u M ) = ( x x. M ) )
9	6, 7, 8	syl2anr 290	. . . . . 6 |- ( ( M e. ZZ /\ x e. ZZ ) -> ( -u x x. -u M ) = ( x x. M ) )
10	9	adantlr 477	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( -u x x. -u M ) = ( x x. M ) )
11	10	eqeq1d 2243	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( -u x x. -u M ) = N <-> ( x x. M ) = N ) )
12	11	biimprd 158	. . 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 12518	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> -u M || N ) )
14		mulneg12 8689	. . . . . . 7 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. M ) = ( x x. -u M ) )
15	6, 7, 14	syl2anr 290	. . . . . 6 |- ( ( M e. ZZ /\ x e. ZZ ) -> ( -u x x. M ) = ( x x. -u M ) )
16	15	adantlr 477	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( -u x x. M ) = ( x x. -u M ) )
17	16	eqeq1d 2243	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( -u x x. M ) = N <-> ( x x. -u M ) = N ) )
18	17	biimprd 158	. . 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 12518	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M || N -> M || N ) )
20	13, 19	impbid 129	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> -u M || N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   class class class wbr 4115  (class class class)co 6059   CCcc 8142   x. cmul 8149   -ucneg 8463   ZZcz 9598   || cdvds 12503

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-0id 8252  ax-rnegex 8253  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-ltadd 8260

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-iota 5318  df-fun 5360  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-inn 9259  df-z 9599  df-dvds 12504

This theorem is referenced by:  absdvdsb  12525  zdvdsdc  12528  3dvds  12580  bezoutlemzz  12728  lcmneg  12801