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Theorem tvclvec 21942
Description: A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tvclvec (𝑊 ∈ TopVec → 𝑊 ∈ LVec)

Proof of Theorem tvclvec
StepHypRef Expression
1 tvclmod 21941 . 2 (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
2 eqid 2621 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
32tvctdrg 21936 . . 3 (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ TopDRing)
4 tdrgdrng 21917 . . 3 ((Scalar‘𝑊) ∈ TopDRing → (Scalar‘𝑊) ∈ DivRing)
53, 4syl 17 . 2 (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ DivRing)
62islvec 19044 . 2 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing))
71, 5, 6sylanbrc 697 1 (𝑊 ∈ TopVec → 𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cfv 5857  Scalarcsca 15884  DivRingcdr 18687  LModclmod 18803  LVecclvec 19042  TopDRingctdrg 21900  TopVecctvc 21902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-lvec 19043  df-tdrg 21904  df-tlm 21905  df-tvc 21906
This theorem is referenced by: (None)
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