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Theorem dvds0lem 12862
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 6090 . . . . . . . . 9  |-  ( x  =  K  ->  (
x  x.  M )  =  ( K  x.  M ) )
21eqeq1d 2446 . . . . . . . 8  |-  ( x  =  K  ->  (
( x  x.  M
)  =  N  <->  ( K  x.  M )  =  N ) )
32rspcev 3054 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  ( K  x.  M
)  =  N )  ->  E. x  e.  ZZ  ( x  x.  M
)  =  N )
43adantl 454 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( K  x.  M )  =  N ) )  ->  E. x  e.  ZZ  ( x  x.  M
)  =  N )
5 divides 12856 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. x  e.  ZZ  (
x  x.  M )  =  N ) )
65adantr 453 . . . . . 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 225 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( K  x.  M )  =  N ) )  ->  M  ||  N )
87expr 600 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( K  x.  M )  =  N  ->  M  ||  N
) )
983impa 1149 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  x.  M
)  =  N  ->  M  ||  N ) )
1093comr 1162 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  =  N  ->  M  ||  N ) )
1110imp 420 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214  (class class class)co 6083    x. cmul 8997   ZZcz 10284    || cdivides 12854
This theorem is referenced by:  iddvds  12865  1dvds  12866  dvds0  12867  dvdsmul1  12873  dvdsmul2  12874  divalgmod  12928  isprm5  13114  dvdsdivcl  20968  ex-dvds  21758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-iota 5420  df-fv 5464  df-ov 6086  df-dvds 12855
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