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Theorem dvds0lem 11708
Description: A lemma to assist theorems of  || with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvds0lem  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  x.  M
)  =  N )  ->  M  ||  N
)

Proof of Theorem dvds0lem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq1 5833 . . . . . . . . 9  |-  ( x  =  K  ->  (
x  x.  M )  =  ( K  x.  M ) )
21eqeq1d 2166 . . . . . . . 8  |-  ( x  =  K  ->  (
( x  x.  M
)  =  N  <->  ( K  x.  M )  =  N ) )
32rspcev 2816 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  ( K  x.  M
)  =  N )  ->  E. x  e.  ZZ  ( x  x.  M
)  =  N )
43adantl 275 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( K  x.  M )  =  N ) )  ->  E. x  e.  ZZ  ( x  x.  M
)  =  N )
5 divides 11696 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. x  e.  ZZ  (
x  x.  M )  =  N ) )
65adantr 274 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( K  x.  M )  =  N ) )  -> 
( M  ||  N  <->  E. x  e.  ZZ  (
x  x.  M )  =  N ) )
74, 6mpbird 166 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( K  x.  M )  =  N ) )  ->  M  ||  N )
87expr 373 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( K  x.  M )  =  N  ->  M  ||  N
) )
983impa 1177 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  x.  M
)  =  N  ->  M  ||  N ) )
1093comr 1193 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  =  N  ->  M  ||  N ) )
1110imp 123 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    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   E.wrex 2436   class class class wbr 3967  (class class class)co 5826    x. cmul 7739   ZZcz 9172    || cdvds 11694
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-iota 5137  df-fv 5180  df-ov 5829  df-dvds 11695
This theorem is referenced by:  iddvds  11711  1dvds  11712  dvds0  11713  dvdsmul1  11720  dvdsmul2  11721  divalgmod  11830  oddpwdclemxy  12059  ex-dvds  13377
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