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Theorem islvec 19085
 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 6178 . . . 4 (𝑓 = 𝑊 → (Scalar‘𝑓) = (Scalar‘𝑊))
2 islvec.1 . . . 4 𝐹 = (Scalar‘𝑊)
31, 2syl6eqr 2672 . . 3 (𝑓 = 𝑊 → (Scalar‘𝑓) = 𝐹)
43eleq1d 2684 . 2 (𝑓 = 𝑊 → ((Scalar‘𝑓) ∈ DivRing ↔ 𝐹 ∈ DivRing))
5 df-lvec 19084 . 2 LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
64, 5elrab2 3360 1 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ‘cfv 5876  Scalarcsca 15925  DivRingcdr 18728  LModclmod 18844  LVecclvec 19083 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-lvec 19084 This theorem is referenced by:  lvecdrng  19086  lveclmod  19087  lsslvec  19088  lvecprop2d  19147  lvecpropd  19148  rlmlvec  19187  frlmphl  20101  tvclvec  21983  isnvc2  22484  iscvs  22908  cnstrcvs  22922  zclmncvs  22929  lindsdom  33374  lindsenlbs  33375  lduallvec  34260  dvalveclem  36133  dvhlveclem  36216  lmod1zrnlvec  42048  aacllem  42312
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