Theorem dvdssub2 11702

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 9187	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
2	1	3adant1 1000	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
3		dvds2sub 11695	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ ( M - N ) e. ZZ ) -> ( ( K || M /\ K || ( M - N ) ) -> K || ( M - ( M - N ) ) ) )
4	2, 3	syld3an3 1262	. . . . . 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 9151	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		zcn 9151	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
9		nncan 8083	. . . . . . 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 1000	. . . . 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 3986	. . 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 11694	. . . . . 6 |- ( ( K e. ZZ /\ ( M - N ) e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || N ) -> K || ( ( M - N ) + N ) ) )
16	2, 15	syld3an2 1264	. . . . 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 8063	. . . . . . 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 1000	. . . . 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 3986	. . 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 963   = wceq 1332   e. wcel 2125   class class class wbr 3961  (class class class)co 5814   CCcc 7709   + caddc 7714   - cmin 8025   ZZcz 9146   || cdvds 11660

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-ltadd 7827

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-iota 5128  df-fun 5165  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-inn 8813  df-n0 9070  df-z 9147  df-dvds 11661

This theorem is referenced by:  dvdsadd  11703