Theorem negdvdsb 11841

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 9309	. . . 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 9309	. . . 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 9283	. . . . . . 7 |- ( x e. ZZ -> x e. CC )
7		zcn 9283	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		mul2neg 8380	. . . . . . 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 2198	. . . 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 11836	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> -u M || N ) )
14		mulneg12 8379	. . . . . . 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 2198	. . . 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 11836	. 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 1364   e. wcel 2160   class class class wbr 4018  (class class class)co 5892   CCcc 7834   x. cmul 7841   -ucneg 8154   ZZcz 9278   || cdvds 11821

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-ltadd 7952

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5234  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-inn 8945  df-z 9279  df-dvds 11822

This theorem is referenced by:  absdvdsb  11843  zdvdsdc  11846  bezoutlemzz  12030  lcmneg  12101