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Theorem islvec 17945
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 5848 . . . 4  |-  ( f  =  W  ->  (Scalar `  f )  =  (Scalar `  W ) )
2 islvec.1 . . . 4  |-  F  =  (Scalar `  W )
31, 2syl6eqr 2513 . . 3  |-  ( f  =  W  ->  (Scalar `  f )  =  F )
43eleq1d 2523 . 2  |-  ( f  =  W  ->  (
(Scalar `  f )  e.  DivRing 
<->  F  e.  DivRing ) )
5 df-lvec 17944 . 2  |-  LVec  =  { f  e.  LMod  |  (Scalar `  f )  e.  DivRing }
64, 5elrab2 3256 1  |-  ( W  e.  LVec  <->  ( W  e. 
LMod  /\  F  e.  DivRing ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   ` cfv 5570  Scalarcsca 14787   DivRingcdr 17591   LModclmod 17707   LVecclvec 17943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-lvec 17944
This theorem is referenced by:  lvecdrng  17946  lveclmod  17947  lsslvec  17948  lvecprop2d  18007  lvecpropd  18008  rlmlvec  18047  frlmphl  18983  tvclvec  20867  isnvc2  21373  lmod1zrnlvec  33349  aacllem  33604  lduallvec  35276  dvalveclem  37149  dvhlveclem  37232
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