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Theorem dvds0lem 12539
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 5865 . . . . . . . . 9  |-  ( x  =  K  ->  (
x  x.  M )  =  ( K  x.  M ) )
21eqeq1d 2291 . . . . . . . 8  |-  ( x  =  K  ->  (
( x  x.  M
)  =  N  <->  ( K  x.  M )  =  N ) )
32rspcev 2884 . . . . . . 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 12533 . . . . . . 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 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023  (class class class)co 5858    x. cmul 8742   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  iddvds  12542  1dvds  12543  dvds0  12544  dvdsmul1  12550  dvdsmul2  12551  divalgmod  12605  isprm5  12791  dvdsdivcl  20421  ex-dvds  20835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861  df-dvds 12532
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