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Theorem dvds2add 12009
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 9385 . . . 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 9385 . . 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 9350 . . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9 zcn 9350 . . . . . . . 8 |- ( y e. ZZ -> y e. CC )
10 zcn 9350 . . . . . . . 8 |- ( K e. ZZ -> K e. CC )
11 adddir 8036 . . . . . . . 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 5934 . . . . 5 |- ( ( ( x x. K ) = M /\ ( y x. K ) = N ) -> ( ( x x. K ) + ( y x. K ) ) = ( M + N ) )
1614, 15sylan9eq 2249 . . . 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 11987 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 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7896   + caddc 7901   x. cmul 7903   ZZcz 9345   || cdvds 11971
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-ltadd 8014
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-inn 9010  df-n0 9269  df-z 9346  df-dvds 11972
This theorem is referenced by:  dvds2addd  12013  dvdssub2  12019  dvdsadd2b  12024  bezoutlemstep  12191  bezoutlembi  12199  dvdsmulgcd  12219  bezoutr  12226  pythagtriplem19  12478  4sqlem16  12602  dec2dvds  12607  lgsquadlem1  15426
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