Theorem dvdsnegb 12119

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 9403	. . . 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 9403	. . . 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 9377	. . . . 5 |- ( x e. ZZ -> x e. CC )
7		zcn 9377	. . . . 5 |- ( M e. ZZ -> M e. CC )
8		mulneg1 8467	. . . . . 6 |- ( ( x e. CC /\ M e. CC ) -> ( -u x x. M ) = -u ( x x. M ) )
9		negeq 8265	. . . . . . 7 |- ( ( x x. M ) = N -> -u ( x x. M ) = -u N )
10	9	eqeq2d 2217	. . . . . 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 12113	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> M || -u N ) )
15		zcn 9377	. . . . . 6 |- ( N e. ZZ -> N e. CC )
16		negeq 8265	. . . . . . . . . 10 |- ( ( x x. M ) = -u N -> -u ( x x. M ) = -u -u N )
17		negneg 8322	. . . . . . . . . 10 |- ( N e. CC -> -u -u N = N )
18	16, 17	sylan9eqr 2260	. . . . . . . . 9 |- ( ( N e. CC /\ ( x x. M ) = -u N ) -> -u ( x x. M ) = N )
19	8, 18	sylan9eq 2258	. . . . . . . 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 1197	. . . . . 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 1292	. . . . 5 |- ( ( x e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
23	22	3coml 1213	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) = -u N -> ( -u x x. M ) = N ) )
24	23	3expa 1206	. . 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 12113	. 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 1373   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   CCcc 7923   x. cmul 7930   -ucneg 8244   ZZcz 9372   || cdvds 12098

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-z 9373  df-dvds 12099

This theorem is referenced by:  dvdsabsb  12121  dvdssub  12149  dvdsadd2b  12151  3dvds  12175  bitscmp  12269  gcdneg  12303  bezoutlemaz  12324  bezoutlembz  12325  prmdiv  12557  pcneg  12648  znunit  14421