MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  islvec Structured version   Unicode version

Theorem islvec 16207
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 5757 . . . 4  |-  ( f  =  W  ->  (Scalar `  f )  =  (Scalar `  W ) )
2 islvec.1 . . . 4  |-  F  =  (Scalar `  W )
31, 2syl6eqr 2492 . . 3  |-  ( f  =  W  ->  (Scalar `  f )  =  F )
43eleq1d 2508 . 2  |-  ( f  =  W  ->  (
(Scalar `  f )  e.  DivRing 
<->  F  e.  DivRing ) )
5 df-lvec 16206 . 2  |-  LVec  =  { f  e.  LMod  |  (Scalar `  f )  e.  DivRing }
64, 5elrab2 3100 1  |-  ( W  e.  LVec  <->  ( W  e. 
LMod  /\  F  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   ` cfv 5483  Scalarcsca 13563   DivRingcdr 15866   LModclmod 15981   LVecclvec 16205
This theorem is referenced by:  lvecdrng  16208  lveclmod  16209  lsslvec  16210  lvecprop2d  16269  lvecpropd  16270  rlmlvec  16308  tvclvec  18259  isnvc2  18765  lduallvec  30050  dvalveclem  31921  dvhlveclem  32004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-lvec 16206
  Copyright terms: Public domain W3C validator