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Theorem muldvds2 11087
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 8793 . . . 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 1138 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  N  e.  ZZ )
)
4 3simpc 942 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
5 zmulcl 8793 . . . 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 1105 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  K )  e.  ZZ )
8 zcn 8745 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 8745 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
10 zcn 8745 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
11 mulass 7463 . . . . . . . 8  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
128, 9, 10, 11syl3an 1216 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
13123coml 1150 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
14133expa 1143 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  K )  x.  M )  =  ( x  x.  ( K  x.  M ) ) )
15143adantl3 1101 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  K )  x.  M )  =  ( x  x.  ( K  x.  M ) ) )
1615eqeq1d 2096 . . 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 11072 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 924    = wceq 1289    e. wcel 1438   class class class wbr 3843  (class class class)co 5644   CCcc 7338    x. cmul 7345   ZZcz 8740    || cdvds 11061
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 3955  ax-pow 4007  ax-pr 4034  ax-setind 4351  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-mulrcl 7434  ax-addcom 7435  ax-mulcom 7436  ax-addass 7437  ax-mulass 7438  ax-distr 7439  ax-i2m1 7440  ax-1rid 7442  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446
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 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-iota 4975  df-fun 5012  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-sub 7645  df-neg 7646  df-inn 8413  df-n0 8664  df-z 8741  df-dvds 11062
This theorem is referenced by: (None)
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