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Theorem muldvds2 11526
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 9114 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
21anim1i 338 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  e.  ZZ  /\  N  e.  ZZ ) )
323impa 1176 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  N  e.  ZZ )
)
4 3simpc 980 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
5 zmulcl 9114 . . . 4  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ )  ->  ( x  x.  K
)  e.  ZZ )
65ancoms 266 . . 3  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ )  ->  ( x  x.  K
)  e.  ZZ )
763ad2antl1 1143 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  K )  e.  ZZ )
8 zcn 9066 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 9066 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
10 zcn 9066 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
11 mulass 7758 . . . . . . . 8  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
128, 9, 10, 11syl3an 1258 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
13123coml 1188 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
14133expa 1181 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  K )  x.  M )  =  ( x  x.  ( K  x.  M ) ) )
15143adantl3 1139 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  K )  x.  M )  =  ( x  x.  ( K  x.  M ) ) )
1615eqeq1d 2148 . . 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 157 . 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 11511 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7625    x. cmul 7632   ZZcz 9061    || cdvds 11500
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-dvds 11501
This theorem is referenced by: (None)
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