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Theorem tvclvec 21812
 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 21811 . 2 (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
2 eqid 2610 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
32tvctdrg 21806 . . 3 (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ TopDRing)
4 tdrgdrng 21787 . . 3 ((Scalar‘𝑊) ∈ TopDRing → (Scalar‘𝑊) ∈ DivRing)
53, 4syl 17 . 2 (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ DivRing)
62islvec 18925 . 2 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing))
71, 5, 6sylanbrc 695 1 (𝑊 ∈ TopVec → 𝑊 ∈ LVec)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ‘cfv 5804  Scalarcsca 15771  DivRingcdr 18570  LModclmod 18686  LVecclvec 18923  TopDRingctdrg 21770  TopVecctvc 21772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-lvec 18924  df-tdrg 21774  df-tlm 21775  df-tvc 21776 This theorem is referenced by: (None)
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