Theorem dvdsnegb 11770

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 9243	. . . 4 |- ( N e. ZZ -> -u N e. ZZ )
3	2	anim2i 340	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ -u N e. ZZ ) )
4		znegcl 9243	. . . 4 |- ( x e. ZZ -> -u x e. ZZ )
5	4	adantl 275	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> -u x e. ZZ )
6		zcn 9217	. . . . 5 |- ( x e. ZZ -> x e. CC )
7		zcn 9217	. . . . 5 |- ( M e. ZZ -> M e. CC )
8		mulneg1 8314	. . . . . 6 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. M ) = -u ( x x. M ) )
9		negeq 8112	. . . . . . 7 |- ( ( x x. M ) = N -> -u ( x x. M ) = -u N )
10	9	eqeq2d 2182	. . . . . 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 288	. . . 4 |- ( ( M e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( -u x x. M ) = -u N ) )
13	12	adantlr 474	. . 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 11764	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> M || -u N ) )
15		zcn 9217	. . . . . 6 |- ( N e. ZZ -> N e. CC )
16		negeq 8112	. . . . . . . . . 10 |- ( ( x x. M ) = -u N -> -u ( x x. M ) = -u -u N )
17		negneg 8169	. . . . . . . . . 10 |- ( N e. CC -> -u -u N = N )
18	16, 17	sylan9eqr 2225	. . . . . . . . 9 |- ( ( N e. CC /\ ( x x. M ) = -u N ) -> -u ( x x. M ) = N )
19	8, 18	sylan9eq 2223	. . . . . . . 8 |- ( ( ( x e. CC /\ M e. CC ) /\ ( N e. CC /\ ( x x. M ) = -u N ) ) -> ( -u x x. M ) = N )
20	19	expr 373	. . . . . . 7 |- ( ( ( x e. CC /\ M e. CC ) /\ N e. CC ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
21	20	3impa 1189	. . . . . 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 1275	. . . . 5 |- ( ( x e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
23	22	3coml 1205	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
24	23	3expa 1198	. . 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 11764	. 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 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   x. cmul 7779   -ucneg 8091   ZZcz 9212   || cdvds 11749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-z 9213  df-dvds 11750

This theorem is referenced by:  dvdsabsb  11772  dvdssub  11800  dvdsadd2b  11802  gcdneg  11937  bezoutlemaz  11958  bezoutlembz  11959  prmdiv  12189  pcneg  12278