Theorem dvdssub2 12261

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 9448	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
2	1	3adant1 1018	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
3		dvds2sub 12252	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ ( M - N ) e. ZZ ) -> ( ( K || M /\ K || ( M - N ) ) -> K || ( M - ( M - N ) ) ) )
4	2, 3	syld3an3 1295	. . . . . 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 9412	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
8		zcn 9412	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
9		nncan 8336	. . . . . . 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 1018	. . . . 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 4085	. . 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 12251	. . . . . 6 |- ( ( K e. ZZ /\ ( M - N ) e. ZZ /\ N e. ZZ ) -> ( ( K || ( M - N ) /\ K || N ) -> K || ( ( M - N ) + N ) ) )
16	2, 15	syld3an2 1297	. . . . 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 8316	. . . . . . 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 1018	. . . . 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 4085	. . 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 981   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   CCcc 7958   + caddc 7963   - cmin 8278   ZZcz 9407   || cdvds 12213

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-dvds 12214

This theorem is referenced by:  dvdsadd  12262  3dvds  12290  bitsmod  12382  bitsinv1lem  12387  znunit  14536  perfectlem1  15586  2sqlem8  15715