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Theorem dvds2add 11827
Description: If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvds2add  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  +  N )
) )

Proof of Theorem dvds2add
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpa 994 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  M  e.  ZZ ) )
2 3simpb 995 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  N  e.  ZZ ) )
3 zaddcl 9291 . . . 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 1199 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  ( M  +  N )  e.  ZZ ) )
6 zaddcl 9291 . . 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 9256 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 9256 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
10 zcn 9256 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
11 adddir 7947 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  K  e.  CC )  ->  (
( x  +  y )  x.  K )  =  ( ( x  x.  K )  +  ( y  x.  K
) ) )
128, 9, 10, 11syl3an 1280 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ  /\  K  e.  ZZ )  ->  (
( x  +  y )  x.  K )  =  ( ( x  x.  K )  +  ( y  x.  K
) ) )
13123comr 1211 . . . . . 6  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ  /\  y  e.  ZZ )  ->  (
( x  +  y )  x.  K )  =  ( ( x  x.  K )  +  ( y  x.  K
) ) )
14133expb 1204 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  +  y )  x.  K )  =  ( ( x  x.  K )  +  ( y  x.  K ) ) )
15 oveq12 5883 . . . . 5  |-  ( ( ( x  x.  K
)  =  M  /\  ( y  x.  K
)  =  N )  ->  ( ( x  x.  K )  +  ( y  x.  K
) )  =  ( M  +  N ) )
1614, 15sylan9eq 2230 . . . 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 1159 . 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 11805 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 978    = wceq 1353    e. wcel 2148   class class class wbr 4003  (class class class)co 5874   CCcc 7808    + caddc 7813    x. cmul 7815   ZZcz 9251    || cdvds 11789
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-dvds 11790
This theorem is referenced by:  dvds2addd  11831  dvdssub2  11837  dvdsadd2b  11842  bezoutlemstep  11992  bezoutlembi  12000  dvdsmulgcd  12020  bezoutr  12027  pythagtriplem19  12276
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