Theorem dvdsnegb 11358

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 8989	. . . 4 |- ( N e. ZZ -> -u N e. ZZ )
3	2	anim2i 337	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ -u N e. ZZ ) )
4		znegcl 8989	. . . 4 |- ( x e. ZZ -> -u x e. ZZ )
5	4	adantl 273	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> -u x e. ZZ )
6		zcn 8963	. . . . 5 |- ( x e. ZZ -> x e. CC )
7		zcn 8963	. . . . 5 |- ( M e. ZZ -> M e. CC )
8		mulneg1 8076	. . . . . 6 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. M ) = -u ( x x. M ) )
9		negeq 7878	. . . . . . 7 |- ( ( x x. M ) = N -> -u ( x x. M ) = -u N )
10	9	eqeq2d 2126	. . . . . 6 |- ( ( x x. M ) = N -> ( ( -u x x. M ) = -u ( x x. M ) <-> ( -u x x. M ) = -u N ) )
11	8, 10	syl5ibcom 154	. . . . 5 |- ( ( x e. CC /\ M e. CC ) -> ( ( x x. M ) = N -> ( -u x x. M ) = -u N ) )
12	6, 7, 11	syl2anr 286	. . . 4 |- ( ( M e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( -u x x. M ) = -u N ) )
13	12	adantlr 466	. . 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 11352	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> M || -u N ) )
15		zcn 8963	. . . . . 6 |- ( N e. ZZ -> N e. CC )
16		negeq 7878	. . . . . . . . . 10 |- ( ( x x. M ) = -u N -> -u ( x x. M ) = -u -u N )
17		negneg 7935	. . . . . . . . . 10 |- ( N e. CC -> -u -u N = N )
18	16, 17	sylan9eqr 2169	. . . . . . . . 9 |- ( ( N e. CC /\ ( x x. M ) = -u N ) -> -u ( x x. M ) = N )
19	8, 18	sylan9eq 2167	. . . . . . . 8 |- ( ( ( x e. CC /\ M e. CC ) /\ ( N e. CC /\ ( x x. M ) = -u N ) ) -> ( -u x x. M ) = N )
20	19	expr 370	. . . . . . 7 |- ( ( ( x e. CC /\ M e. CC ) /\ N e. CC ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
21	20	3impa 1159	. . . . . 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 1241	. . . . 5 |- ( ( x e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
23	22	3coml 1171	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
24	23	3expa 1164	. . 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 11352	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || -u N -> M || N ) )
26	14, 25	impbid 128	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || -u N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1314   e. wcel 1463   class class class wbr 3895  (class class class)co 5728   CCcc 7545   x. cmul 7552   -ucneg 7857   ZZcz 8958   || cdvds 11341

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661

This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-z 8959  df-dvds 11342

This theorem is referenced by:  dvdsabsb  11360  dvdssub  11386  dvdsadd2b  11388  gcdneg  11518  bezoutlemaz  11537  bezoutlembz  11538