Theorem dvdssub2 11978

Description: If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)

Assertion

Ref	Expression
dvdssub2	|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) -> ( K || M <-> K || N ) )

Proof of Theorem dvdssub2

Step	Hyp	Ref	Expression
1		zsubcl 9358	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
2	1	3adant1 1017	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
3		dvds2sub 11969	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ ( M - N ) e. ZZ ) -> ( ( K || M /\ K || ( M - N ) ) -> K || ( M - ( M - N ) ) ) )
4	2, 3	syld3an3 1294	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || ( M - N ) ) -> K || ( M - ( M - N ) ) ) )
5	4	ancomsd 269	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || M ) -> K || ( M - ( M - N ) ) ) )
6	5	imp 124	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || M ) ) -> K || ( M - ( M - N ) ) )
7		zcn 9322	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		zcn 9322	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
9		nncan 8248	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( M - ( M - N ) ) = N )
10	7, 8, 9	syl2an 289	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - ( M - N ) ) = N )
11	10	3adant1 1017	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M - ( M - N ) ) = N )
12	11	adantr 276	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || M ) ) -> ( M - ( M - N ) ) = N )
13	6, 12	breqtrd 4055	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || M ) ) -> K || N )
14	13	expr 375	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) -> ( K || M -> K || N ) )
15		dvds2add 11968	. . . . . 6 |- ( ( K e. ZZ /\ ( M - N ) e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || N ) -> K || ( ( M - N ) + N ) ) )
16	2, 15	syld3an2 1296	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || N ) -> K || ( ( M - N ) + N ) ) )
17	16	imp 124	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || N ) ) -> K || ( ( M - N ) + N ) )
18		npcan 8228	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M - N ) + N ) = M )
19	7, 8, 18	syl2an 289	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M - N ) + N ) = M )
20	19	3adant1 1017	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M - N ) + N ) = M )
21	20	adantr 276	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || N ) ) -> ( ( M - N ) + N ) = M )
22	17, 21	breqtrd 4055	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || N ) ) -> K || M )
23	22	expr 375	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) -> ( K || N -> K || M ) )
24	14, 23	impbid 129	1 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) -> ( K || M <-> K || N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   CCcc 7870   + caddc 7875   - cmin 8190   ZZcz 9317   || cdvds 11930

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-dvds 11931

This theorem is referenced by:  dvdsadd  11979  znunit  14147  2sqlem8  15210