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Theorem islvec 16135
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 5691 . . . 4  |-  ( f  =  W  ->  (Scalar `  f )  =  (Scalar `  W ) )
2 islvec.1 . . . 4  |-  F  =  (Scalar `  W )
31, 2syl6eqr 2458 . . 3  |-  ( f  =  W  ->  (Scalar `  f )  =  F )
43eleq1d 2474 . 2  |-  ( f  =  W  ->  (
(Scalar `  f )  e.  DivRing 
<->  F  e.  DivRing ) )
5 df-lvec 16134 . 2  |-  LVec  =  { f  e.  LMod  |  (Scalar `  f )  e.  DivRing }
64, 5elrab2 3058 1  |-  ( W  e.  LVec  <->  ( W  e. 
LMod  /\  F  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5417  Scalarcsca 13491   DivRingcdr 15794   LModclmod 15909   LVecclvec 16133
This theorem is referenced by:  lvecdrng  16136  lveclmod  16137  lsslvec  16138  lvecprop2d  16197  lvecpropd  16198  rlmlvec  16236  tvclvec  18185  isnvc2  18691  lduallvec  29641  dvalveclem  31512  dvhlveclem  31595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-iota 5381  df-fv 5425  df-lvec 16134
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