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Theorem muldvds2 10366
Description: If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
muldvds2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  ||  N  ->  M 
||  N ) )

Proof of Theorem muldvds2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 zmulcl 8485 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
21anim1i 333 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  e.  ZZ  /\  N  e.  ZZ ) )
323impa 1134 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  N  e.  ZZ )
)
4 3simpc 938 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
5 zmulcl 8485 . . . 4  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ )  ->  ( x  x.  K
)  e.  ZZ )
65ancoms 264 . . 3  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ )  ->  ( x  x.  K
)  e.  ZZ )
763ad2antl1 1101 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  K )  e.  ZZ )
8 zcn 8437 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 8437 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
10 zcn 8437 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
11 mulass 7166 . . . . . . . 8  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
128, 9, 10, 11syl3an 1212 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
13123coml 1146 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
14133expa 1139 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  K )  x.  M )  =  ( x  x.  ( K  x.  M ) ) )
15143adantl3 1097 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  K )  x.  M )  =  ( x  x.  ( K  x.  M ) ) )
1615eqeq1d 2090 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( ( x  x.  K )  x.  M )  =  N  <->  ( x  x.  ( K  x.  M
) )  =  N ) )
1716biimprd 156 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  ( K  x.  M ) )  =  N  ->  ( (
x  x.  K )  x.  M )  =  N ) )
183, 4, 7, 17dvds1lem 10351 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  ||  N  ->  M 
||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041    x. cmul 7048   ZZcz 8432    || cdvds 10340
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-dvds 10341
This theorem is referenced by: (None)
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