Theorem dvdsnegb 12519

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 9625	. . . 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 9625	. . . 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 9599	. . . . 5 |- ( x e. ZZ -> x e. CC )
7		zcn 9599	. . . . 5 |- ( M e. ZZ -> M e. CC )
8		mulneg1 8685	. . . . . 6 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. M ) = -u ( x x. M ) )
9		negeq 8482	. . . . . . 7 |- ( ( x x. M ) = N -> -u ( x x. M ) = -u N )
10	9	eqeq2d 2246	. . . . . 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 12513	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> M || -u N ) )
15		zcn 9599	. . . . . 6 |- ( N e. ZZ -> N e. CC )
16		negeq 8482	. . . . . . . . . 10 |- ( ( x x. M ) = -u N -> -u ( x x. M ) = -u -u N )
17		negneg 8539	. . . . . . . . . 10 |- ( N e. CC -> -u -u N = N )
18	16, 17	sylan9eqr 2289	. . . . . . . . 9 |- ( ( N e. CC /\ ( x x. M ) = -u N ) -> -u ( x x. M ) = N )
19	8, 18	sylan9eq 2287	. . . . . . . 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 12513	. 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 2205   class class class wbr 4114  (class class class)co 6058   CCcc 8141   x. cmul 8148   -ucneg 8461   ZZcz 9594   || cdvds 12498

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-z 9595  df-dvds 12499

This theorem is referenced by:  dvdsabsb  12521  dvdssub  12549  dvdsadd2b  12551  3dvds  12575  bitscmp  12669  gcdneg  12703  bezoutlemaz  12724  bezoutlembz  12725  prmdiv  12957  pcneg  13048  znunit  14933