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Theorem dvds0lem 12555
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 5881 . . . . . . . . 9  |-  ( x  =  K  ->  (
x  x.  M )  =  ( K  x.  M ) )
21eqeq1d 2304 . . . . . . . 8  |-  ( x  =  K  ->  (
( x  x.  M
)  =  N  <->  ( K  x.  M )  =  N ) )
32rspcev 2897 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  ( K  x.  M
)  =  N )  ->  E. x  e.  ZZ  ( x  x.  M
)  =  N )
43adantl 452 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( K  x.  M )  =  N ) )  ->  E. x  e.  ZZ  ( x  x.  M
)  =  N )
5 divides 12549 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. x  e.  ZZ  (
x  x.  M )  =  N ) )
65adantr 451 . . . . . 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 223 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( K  x.  M )  =  N ) )  ->  M  ||  N )
87expr 598 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( K  x.  M )  =  N  ->  M  ||  N
) )
983impa 1146 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  x.  M
)  =  N  ->  M  ||  N ) )
1093comr 1159 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  =  N  ->  M  ||  N ) )
1110imp 418 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039  (class class class)co 5874    x. cmul 8758   ZZcz 10040    || cdivides 12547
This theorem is referenced by:  iddvds  12558  1dvds  12559  dvds0  12560  dvdsmul1  12566  dvdsmul2  12567  divalgmod  12621  isprm5  12807  dvdsdivcl  20437  ex-dvds  20851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-iota 5235  df-fv 5279  df-ov 5877  df-dvds 12548
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