Theorem dvds2sub 11517

Description: If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
dvds2sub	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M - N ) ) )

Proof of Theorem dvds2sub

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpa 978	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ M e. ZZ ) )
2		3simpb 979	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
3		zsubcl 9088	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
4	3	anim2i 339	. . 3 |- ( ( K e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( K e. ZZ /\ ( M - N ) e. ZZ ) )
5	4	3impb 1177	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ ( M - N ) e. ZZ ) )
6		zsubcl 9088	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x - y ) e. ZZ )
7	6	adantl 275	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x - y ) e. ZZ )
8		zcn 9052	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9		zcn 9052	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
10		zcn 9052	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
11		subdir 8141	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
12	8, 9, 10, 11	syl3an 1258	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ /\ K e. ZZ ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
13	12	3comr 1189	. . . . . 6 |- ( ( K e. ZZ /\ x e. ZZ /\ y e. ZZ ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
14	13	3expb 1182	. . . . 5 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
15		oveq12 5776	. . . . 5 |- ( ( ( x x. K ) = M /\ ( y x. K ) = N ) -> ( ( x x. K ) - ( y x. K ) ) = ( M - N ) )
16	14, 15	sylan9eq 2190	. . . 4 |- ( ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) /\ ( ( x x. K ) = M /\ ( y x. K ) = N ) ) -> ( ( x - y ) x. K ) = ( M - N ) )
17	16	ex 114	. . 3 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. K ) = M /\ ( y x. K ) = N ) -> ( ( x - y ) x. K ) = ( M - N ) ) )
18	17	3ad2antl1 1143	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. K ) = M /\ ( y x. K ) = N ) -> ( ( x - y ) x. K ) = ( M - N ) ) )
19	1, 2, 5, 7, 18	dvds2lem 11494	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M - N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   CCcc 7611   x. cmul 7618   - cmin 7926   ZZcz 9047   || cdvds 11482

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-dvds 11483

This theorem is referenced by:  dvds2subd  11518  dvdssub2  11524