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Theorem islvec 21103
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 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))

Proof of Theorem islvec
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . . 4 (𝑓 = 𝑊 → (Scalar‘𝑓) = (Scalar‘𝑊))
2 islvec.1 . . . 4 𝐹 = (Scalar‘𝑊)
31, 2eqtr4di 2795 . . 3 (𝑓 = 𝑊 → (Scalar‘𝑓) = 𝐹)
43eleq1d 2826 . 2 (𝑓 = 𝑊 → ((Scalar‘𝑓) ∈ DivRing ↔ 𝐹 ∈ DivRing))
5 df-lvec 21102 . 2 LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
64, 5elrab2 3695 1 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  cfv 6561  Scalarcsca 17300  DivRingcdr 20729  LModclmod 20858  LVecclvec 21101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-lvec 21102
This theorem is referenced by:  lvecdrng  21104  lveclmod  21105  lsslvec  21108  lmhmlvec  21109  lvecprop2d  21168  lvecpropd  21169  rlmlvec  21211  frlmlvec  21781  frlmphl  21801  mpllvec  22040  tvclvec  24207  isnvc2  24720  iscvs  25160  cnstrcvs  25174  zclmncvs  25182  quslvec  33388  ply1lvec  33585  sralvec  33636  matdim  33666  lmhmlvec2  33670  assalactf1o  33686  ccfldsrarelvec  33721  fldextrspunlem1  33725  fldextrspunfld  33726  bj-isvec  37288  lindsdom  37621  lindsenlbs  37622  lduallvec  39155  dvalveclem  41027  dvhlveclem  41110  lmod1zrnlvec  48411  aacllem  49320
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