ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvdscmul Unicode version

Theorem dvdscmul 11427
Description: Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdscmul  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N
) ) )

Proof of Theorem dvdscmul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 3simpc 963 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 zmulcl 9061 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
323adant3 984 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
4 zmulcl 9061 . . . . 5  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
543adant2 983 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
63, 5jca 302 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
7 simpr 109 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
8 zcn 9013 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 9013 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
10 zcn 9013 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
11 mul12 7855 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
x  x.  ( K  x.  M ) )  =  ( K  x.  ( x  x.  M
) ) )
128, 9, 10, 11syl3an 1241 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
x  x.  ( K  x.  M ) )  =  ( K  x.  ( x  x.  M
) ) )
13123coml 1171 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  x  e.  ZZ )  ->  (
x  x.  ( K  x.  M ) )  =  ( K  x.  ( x  x.  M
) ) )
14133expa 1164 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  ( K  x.  M
) )  =  ( K  x.  ( x  x.  M ) ) )
15143adantl3 1122 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  ( K  x.  M
) )  =  ( K  x.  ( x  x.  M ) ) )
16 oveq2 5748 . . . . 5  |-  ( ( x  x.  M )  =  N  ->  ( K  x.  ( x  x.  M ) )  =  ( K  x.  N
) )
1715, 16sylan9eq 2168 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  /\  (
x  x.  M )  =  N )  -> 
( x  x.  ( K  x.  M )
)  =  ( K  x.  N ) )
1817ex 114 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( x  x.  ( K  x.  M
) )  =  ( K  x.  N ) ) )
191, 6, 7, 18dvds1lem 11411 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N
) ) )
20193coml 1171 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   CCcc 7582    x. cmul 7589   ZZcz 9008    || cdvds 11400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-dvds 11401
This theorem is referenced by:  dvdscmulr  11429  mulgcd  11611  dvdsmulgcd  11620  rpmulgcd2  11683
  Copyright terms: Public domain W3C validator