Theorem dvds2sub 11105

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 940	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ M e. ZZ ) )
2		3simpb 941	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
3		zsubcl 8789	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
4	3	anim2i 334	. . 3 |- ( ( K e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( K e. ZZ /\ ( M - N ) e. ZZ ) )
5	4	3impb 1139	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ ( M - N ) e. ZZ ) )
6		zsubcl 8789	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x - y ) e. ZZ )
7	6	adantl 271	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x - y ) e. ZZ )
8		zcn 8753	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9		zcn 8753	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
10		zcn 8753	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
11		subdir 7862	. . . . . . . 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 1216	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ /\ K e. ZZ ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
13	12	3comr 1151	. . . . . 6 |- ( ( K e. ZZ /\ x e. ZZ /\ y e. ZZ ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
14	13	3expb 1144	. . . . 5 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x - y ) x. K ) = ( ( x x. K ) - ( y x. K ) ) )
15		oveq12 5661	. . . . 5 |- ( ( ( x x. K ) = M /\ ( y x. K ) = N ) -> ( ( x x. K ) - ( y x. K ) ) = ( M - N ) )
16	14, 15	sylan9eq 2140	. . . 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 113	. . 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 1105	. 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 11082	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M - N ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 924   = wceq 1289   e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7346   x. cmul 7353   - cmin 7651   ZZcz 8748   || cdvds 11070

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-ltadd 7459

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-dvds 11071

This theorem is referenced by:  dvds2subd  11106  dvdssub2  11112