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Theorem dvds2sub 11865
Description: If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvds2sub  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  -  N )
) )

Proof of Theorem dvds2sub
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpa 996 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  M  e.  ZZ ) )
2 3simpb 997 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  N  e.  ZZ ) )
3 zsubcl 9324 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N
)  e.  ZZ )
43anim2i 342 . . 3  |-  ( ( K  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( K  e.  ZZ  /\  ( M  -  N )  e.  ZZ ) )
543impb 1201 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  ( M  -  N )  e.  ZZ ) )
6 zsubcl 9324 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  -  y
)  e.  ZZ )
76adantl 277 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  -  y )  e.  ZZ )
8 zcn 9288 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 9288 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
10 zcn 9288 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
11 subdir 8373 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  K  e.  CC )  ->  (
( x  -  y
)  x.  K )  =  ( ( x  x.  K )  -  ( y  x.  K
) ) )
128, 9, 10, 11syl3an 1291 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ  /\  K  e.  ZZ )  ->  (
( x  -  y
)  x.  K )  =  ( ( x  x.  K )  -  ( y  x.  K
) ) )
13123comr 1213 . . . . . 6  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ  /\  y  e.  ZZ )  ->  (
( x  -  y
)  x.  K )  =  ( ( x  x.  K )  -  ( y  x.  K
) ) )
14133expb 1206 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  -  y )  x.  K )  =  ( ( x  x.  K )  -  (
y  x.  K ) ) )
15 oveq12 5905 . . . . 5  |-  ( ( ( x  x.  K
)  =  M  /\  ( y  x.  K
)  =  N )  ->  ( ( x  x.  K )  -  ( y  x.  K
) )  =  ( M  -  N ) )
1614, 15sylan9eq 2242 . . . 4  |-  ( ( ( K  e.  ZZ  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  /\  ( ( x  x.  K )  =  M  /\  (
y  x.  K )  =  N ) )  ->  ( ( x  -  y )  x.  K )  =  ( M  -  N ) )
1716ex 115 . . 3  |-  ( ( K  e.  ZZ  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
( x  x.  K
)  =  M  /\  ( y  x.  K
)  =  N )  ->  ( ( x  -  y )  x.  K )  =  ( M  -  N ) ) )
18173ad2antl1 1161 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
( x  x.  K
)  =  M  /\  ( y  x.  K
)  =  N )  ->  ( ( x  -  y )  x.  K )  =  ( M  -  N ) ) )
191, 2, 5, 7, 18dvds2lem 11842 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  -  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5896   CCcc 7839    x. cmul 7846    - cmin 8158   ZZcz 9283    || cdvds 11826
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-ltadd 7957
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-inn 8950  df-n0 9207  df-z 9284  df-dvds 11827
This theorem is referenced by:  dvds2subd  11866  dvdssub2  11874  difsqpwdvds  12370
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