ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvdsmulc Unicode version
Theorem dvdsmulc 12460
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 1023 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
2 zmulcl 9594 . . . . . 6 |- ( ( M e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
323adant2 1043 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
4 zmulcl 9594 . . . . . 6 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
543adant1 1042 . . . . 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 1238 . . 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 9545 . . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
10 zcn 9545 . . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
11 zcn 9545 . . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
12 mulass 8223 . . . . . . . . 9 |- ( ( x e. CC /\ M e. CC /\ K e. CC ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
139, 10, 11, 12syl3an 1316 . . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ K e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
14133com13 1235 . . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
15143expa 1230 . . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
16153adantl3 1182 . . . . 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 6035 . . . . 5 |- ( ( x x. M ) = N -> ( ( x x. M ) x. K ) = ( N x. K ) )
1816, 17sylan9req 2285 . . . 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 12443 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
21203coml 1237 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 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   CCcc 8090   x. cmul 8097   ZZcz 9540   || cdvds 12428
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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-dvds 12429
This theorem is referenced by:  dvdsmulcr  12462  coprmdvds2  12745  mulgcddvds  12746  rpmulgcd2  12747  pcpremul  12946  znrrg  14756  mpodvdsmulf1o  15804
  Copyright terms: Public domain W3C validator