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Theorem dvdscmul 10916
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 942 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 zmulcl 8773 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
323adant3 963 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
4 zmulcl 8773 . . . . 5  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
543adant2 962 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
63, 5jca 300 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
7 simpr 108 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
8 zcn 8725 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 8725 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
10 zcn 8725 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
11 mul12 7590 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
x  x.  ( K  x.  M ) )  =  ( K  x.  ( x  x.  M
) ) )
128, 9, 10, 11syl3an 1216 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
x  x.  ( K  x.  M ) )  =  ( K  x.  ( x  x.  M
) ) )
13123coml 1150 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  x  e.  ZZ )  ->  (
x  x.  ( K  x.  M ) )  =  ( K  x.  ( x  x.  M
) ) )
14133expa 1143 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  ( K  x.  M
) )  =  ( K  x.  ( x  x.  M ) ) )
15143adantl3 1101 . . . . 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 5642 . . . . 5  |-  ( ( x  x.  M )  =  N  ->  ( K  x.  ( x  x.  M ) )  =  ( K  x.  N
) )
1715, 16sylan9eq 2140 . . . 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 113 . . 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 10900 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N
) ) )
20193coml 1150 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 102    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327    x. cmul 7334   ZZcz 8720    || cdvds 10889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-dvds 10890
This theorem is referenced by:  dvdscmulr  10918  mulgcd  11098  dvdsmulgcd  11107  rpmulgcd2  11170
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