Theorem dvdssub2 11775

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 9232	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
2	1	3adant1 1005	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
3		dvds2sub 11766	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ ( M - N ) e. ZZ ) -> ( ( K || M /\ K || ( M - N ) ) -> K || ( M - ( M - N ) ) ) )
4	2, 3	syld3an3 1273	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || ( M - N ) ) -> K || ( M - ( M - N ) ) ) )
5	4	ancomsd 267	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || M ) -> K || ( M - ( M - N ) ) ) )
6	5	imp 123	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || M ) ) -> K || ( M - ( M - N ) ) )
7		zcn 9196	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		zcn 9196	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
9		nncan 8127	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( M - ( M - N ) ) = N )
10	7, 8, 9	syl2an 287	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - ( M - N ) ) = N )
11	10	3adant1 1005	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M - ( M - N ) ) = N )
12	11	adantr 274	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || M ) ) -> ( M - ( M - N ) ) = N )
13	6, 12	breqtrd 4008	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || M ) ) -> K || N )
14	13	expr 373	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) -> ( K || M -> K || N ) )
15		dvds2add 11765	. . . . . 6 |- ( ( K e. ZZ /\ ( M - N ) e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || N ) -> K || ( ( M - N ) + N ) ) )
16	2, 15	syld3an2 1275	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || N ) -> K || ( ( M - N ) + N ) ) )
17	16	imp 123	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || N ) ) -> K || ( ( M - N ) + N ) )
18		npcan 8107	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M - N ) + N ) = M )
19	7, 8, 18	syl2an 287	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M - N ) + N ) = M )
20	19	3adant1 1005	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M - N ) + N ) = M )
21	20	adantr 274	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || N ) ) -> ( ( M - N ) + N ) = M )
22	17, 21	breqtrd 4008	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || ( M - N ) /\ K || N ) ) -> K || M )
23	22	expr 373	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) -> ( K || N -> K || M ) )
24	14, 23	impbid 128	1 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) -> ( K || M <-> K || N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   + caddc 7756   - cmin 8069   ZZcz 9191   || cdvds 11727

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-dvds 11728

This theorem is referenced by:  dvdsadd  11776  2sqlem8  13599