ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvds0lem Unicode version

Theorem dvds0lem 11763
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 5860 . . . . . . . . 9  |-  ( x  =  K  ->  (
x  x.  M )  =  ( K  x.  M ) )
21eqeq1d 2179 . . . . . . . 8  |-  ( x  =  K  ->  (
( x  x.  M
)  =  N  <->  ( K  x.  M )  =  N ) )
32rspcev 2834 . . . . . . 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 11751 . . . . . . 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 1189 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  x.  M
)  =  N  ->  M  ||  N ) )
1093comr 1206 . 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 973    = wceq 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853    x. cmul 7779   ZZcz 9212    || cdvds 11749
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-iota 5160  df-fv 5206  df-ov 5856  df-dvds 11750
This theorem is referenced by:  iddvds  11766  1dvds  11767  dvds0  11768  dvdsmul1  11775  dvdsmul2  11776  divalgmod  11886  oddpwdclemxy  12123  ex-dvds  13765
  Copyright terms: Public domain W3C validator