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Theorem dvds2add 11968
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 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 zaddcl 9357 . . . 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 zaddcl 9357 . . 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 9322 . . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9 zcn 9322 . . . . . . . 8 |- ( y e. ZZ -> y e. CC )
10 zcn 9322 . . . . . . . 8 |- ( K e. ZZ -> K e. CC )
11 adddir 8010 . . . . . . . 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 5927 . . . . 5 |- ( ( ( x x. K ) = M /\ ( y x. K ) = N ) -> ( ( x x. K ) + ( y x. K ) ) = ( M + N ) )
1614, 15sylan9eq 2246 . . . 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 11946 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 2164   class class class wbr 4029  (class class class)co 5918   CCcc 7870   + caddc 7875   x. cmul 7877   ZZcz 9317   || cdvds 11930
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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-dvds 11931
This theorem is referenced by:  dvds2addd  11972  dvdssub2  11978  dvdsadd2b  11983  bezoutlemstep  12134  bezoutlembi  12142  dvdsmulgcd  12162  bezoutr  12169  pythagtriplem19  12420  4sqlem16  12544  lgsquadlem1  15191
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