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Theorem islvec 15857
Description: The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
Hypothesis
Ref Expression
islvec.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
islvec  |-  ( W  e.  LVec  <->  ( W  e. 
LMod  /\  F  e.  DivRing ) )

Proof of Theorem islvec
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( f  =  W  ->  (Scalar `  f )  =  (Scalar `  W ) )
2 islvec.1 . . . 4  |-  F  =  (Scalar `  W )
31, 2syl6eqr 2333 . . 3  |-  ( f  =  W  ->  (Scalar `  f )  =  F )
43eleq1d 2349 . 2  |-  ( f  =  W  ->  (
(Scalar `  f )  e.  DivRing 
<->  F  e.  DivRing ) )
5 df-lvec 15856 . 2  |-  LVec  =  { f  e.  LMod  |  (Scalar `  f )  e.  DivRing }
64, 5elrab2 2925 1  |-  ( W  e.  LVec  <->  ( W  e. 
LMod  /\  F  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  Scalarcsca 13211   DivRingcdr 15512   LModclmod 15627   LVecclvec 15855
This theorem is referenced by:  lvecdrng  15858  lveclmod  15859  lsslvec  15860  lvecprop2d  15919  lvecpropd  15920  rlmlvec  15958  tvclvec  17881  isnvc2  18209  lduallvec  29344  dvalveclem  31215  dvhlveclem  31298
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  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-iota 5219  df-fv 5263  df-lvec 15856
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