Theorem dvdssub2 11845

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 9297	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
2	1	3adant1 1015	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
3		dvds2sub 11836	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ ( M - N ) e. ZZ ) -> ( ( K || M /\ K || ( M - N ) ) -> K || ( M - ( M - N ) ) ) )
4	2, 3	syld3an3 1283	. . . . . 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 9261	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		zcn 9261	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
9		nncan 8189	. . . . . . 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 1015	. . . . 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 4031	. . 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 11835	. . . . . 6 |- ( ( K e. ZZ /\ ( M - N ) e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || N ) -> K || ( ( M - N ) + N ) ) )
16	2, 15	syld3an2 1285	. . . . 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 8169	. . . . . . 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 1015	. . . . 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 4031	. . 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 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5878   CCcc 7812   + caddc 7817   - cmin 8131   ZZcz 9256   || cdvds 11797

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257  df-dvds 11798

This theorem is referenced by:  dvdsadd  11846  2sqlem8  14610