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Theorem dvds2add 12385
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 1020 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ M e. ZZ ) )
2 3simpb 1021 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
3 zaddcl 9518 . . . 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 1225 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ ( M + N ) e. ZZ ) )
6 zaddcl 9518 . . 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 9483 . . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9 zcn 9483 . . . . . . . 8 |- ( y e. ZZ -> y e. CC )
10 zcn 9483 . . . . . . . 8 |- ( K e. ZZ -> K e. CC )
11 adddir 8169 . . . . . . . 8 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( ( x + y ) x. K ) = ( ( x x. K ) + ( y x. K ) ) )
128, 9, 10, 11syl3an 1315 . . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ /\ K e. ZZ ) -> ( ( x + y ) x. K ) = ( ( x x. K ) + ( y x. K ) ) )
13123comr 1237 . . . . . 6 |- ( ( K e. ZZ /\ x e. ZZ /\ y e. ZZ ) -> ( ( x + y ) x. K ) = ( ( x x. K ) + ( y x. K ) ) )
14133expb 1230 . . . . 5 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x + y ) x. K ) = ( ( x x. K ) + ( y x. K ) ) )
15 oveq12 6026 . . . . 5 |- ( ( ( x x. K ) = M /\ ( y x. K ) = N ) -> ( ( x x. K ) + ( y x. K ) ) = ( M + N ) )
1614, 15sylan9eq 2284 . . . 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 1185 . 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 12363 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 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   + caddc 8034   x. cmul 8036   ZZcz 9478   || cdvds 12347
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147
This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-dvds 12348
This theorem is referenced by:  dvds2addd  12389  dvdssub2  12395  dvdsadd2b  12400  bezoutlemstep  12567  bezoutlembi  12575  dvdsmulgcd  12595  bezoutr  12602  pythagtriplem19  12854  4sqlem16  12978  dec2dvds  12983  lgsquadlem1  15805
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