Theorem dvds2sub 12108

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 996	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ M e. ZZ ) )
2		3simpb 997	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
3		zsubcl 9412	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
4	3	anim2i 342	. . 3 |- ( ( K e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( K e. ZZ /\ ( M - N ) e. ZZ ) )
5	4	3impb 1201	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ ( M - N ) e. ZZ ) )
6		zsubcl 9412	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x - y ) e. ZZ )
7	6	adantl 277	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x - y ) e. ZZ )
8		zcn 9376	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9		zcn 9376	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
10		zcn 9376	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
11		subdir 8457	. . . . . . . 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 1291	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ /\ K e. ZZ ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
13	12	3comr 1213	. . . . . 6 |- ( ( K e. ZZ /\ x e. ZZ /\ y e. ZZ ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
14	13	3expb 1206	. . . . 5 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
15		oveq12 5952	. . . . 5 |- ( ( ( x x. K ) = M /\ ( y x. K ) = N ) -> ( ( x x. K ) - ( y x. K ) ) = ( M - N ) )
16	14, 15	sylan9eq 2257	. . . 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 115	. . 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 1161	. 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 12085	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M - N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1372   e. wcel 2175   class class class wbr 4043  (class class class)co 5943   CCcc 7922   x. cmul 7929   - cmin 8242   ZZcz 9371   || cdvds 12069

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-n0 9295  df-z 9372  df-dvds 12070

This theorem is referenced by:  dvds2subd  12109  dvdssub2  12117  difsqpwdvds  12632