Theorem dvdsnegb 12432

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

Assertion

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

Proof of Theorem dvdsnegb

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 9554	. . . 4 |- ( N e. ZZ -> -u N e. ZZ )
3	2	anim2i 342	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ -u N e. ZZ ) )
4		znegcl 9554	. . . 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 9528	. . . . 5 |- ( x e. ZZ -> x e. CC )
7		zcn 9528	. . . . 5 |- ( M e. ZZ -> M e. CC )
8		mulneg1 8616	. . . . . 6 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. M ) = -u ( x x. M ) )
9		negeq 8414	. . . . . . 7 |- ( ( x x. M ) = N -> -u ( x x. M ) = -u N )
10	9	eqeq2d 2243	. . . . . 6 |- ( ( x x. M ) = N -> ( ( -u x x. M ) = -u ( x x. M ) <-> ( -u x x. M ) = -u N ) )
11	8, 10	syl5ibcom 155	. . . . 5 |- ( ( x e. CC /\ M e. CC ) -> ( ( x x. M ) = N -> ( -u x x. M ) = -u N ) )
12	6, 7, 11	syl2anr 290	. . . 4 |- ( ( M e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( -u x x. M ) = -u N ) )
13	12	adantlr 477	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( -u x x. M ) = -u N ) )
14	1, 3, 5, 13	dvds1lem 12426	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> M || -u N ) )
15		zcn 9528	. . . . . 6 |- ( N e. ZZ -> N e. CC )
16		negeq 8414	. . . . . . . . . 10 |- ( ( x x. M ) = -u N -> -u ( x x. M ) = -u -u N )
17		negneg 8471	. . . . . . . . . 10 |- ( N e. CC -> -u -u N = N )
18	16, 17	sylan9eqr 2286	. . . . . . . . 9 |- ( ( N e. CC /\ ( x x. M ) = -u N ) -> -u ( x x. M ) = N )
19	8, 18	sylan9eq 2284	. . . . . . . 8 |- ( ( ( x e. CC /\ M e. CC ) /\ ( N e. CC /\ ( x x. M ) = -u N ) ) -> ( -u x x. M ) = N )
20	19	expr 375	. . . . . . 7 |- ( ( ( x e. CC /\ M e. CC ) /\ N e. CC ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
21	20	3impa 1221	. . . . . 6 |- ( ( x e. CC /\ M e. CC /\ N e. CC ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
22	6, 7, 15, 21	syl3an 1316	. . . . 5 |- ( ( x e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
23	22	3coml 1237	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
24	23	3expa 1230	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
25	3, 1, 5, 24	dvds1lem 12426	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || -u N -> M || N ) )
26	14, 25	impbid 129	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || -u N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   CCcc 8073   x. cmul 8080   -ucneg 8393   ZZcz 9523   || cdvds 12411

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-z 9524  df-dvds 12412

This theorem is referenced by:  dvdsabsb  12434  dvdssub  12462  dvdsadd2b  12464  3dvds  12488  bitscmp  12582  gcdneg  12616  bezoutlemaz  12637  bezoutlembz  12638  prmdiv  12870  pcneg  12961  znunit  14738