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Theorem dvdsmulc 11825
Description: Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdsmulc |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
Proof of Theorem dvdsmulc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 3simpc 996 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
2 zmulcl 9305 . . . . . 6 |- ( ( M e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
323adant2 1016 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
4 zmulcl 9305 . . . . . 6 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
543adant1 1015 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
63, 5jca 306 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
763comr 1211 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
8 simpr 110 . . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
9 zcn 9257 . . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
10 zcn 9257 . . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
11 zcn 9257 . . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
12 mulass 7941 . . . . . . . . 9 |- ( ( x e. CC /\ M e. CC /\ K e. CC ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
139, 10, 11, 12syl3an 1280 . . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ K e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
14133com13 1208 . . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
15143expa 1203 . . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
16153adantl3 1155 . . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
17 oveq1 5881 . . . . 5 |- ( ( x x. M ) = N -> ( ( x x. M ) x. K ) = ( N x. K ) )
1816, 17sylan9req 2231 . . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) /\ ( x x. M ) = N ) -> ( x x. ( M x. K ) ) = ( N x. K ) )
1918ex 115 . . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( x x. ( M x. K ) ) = ( N x. K ) ) )
201, 7, 8, 19dvds1lem 11808 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
21203coml 1210 1 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
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   x. cmul 7815   ZZcz 9252   || cdvds 11793
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-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  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-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921
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-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-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253  df-dvds 11794
This theorem is referenced by:  dvdsmulcr  11827  coprmdvds2  12092  mulgcddvds  12093  rpmulgcd2  12094  pcpremul  12292
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