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Theorem tvclvec 24208
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 24207 . 2 (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
2 eqid 2736 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
32tvctdrg 24202 . . 3 (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ TopDRing)
4 tdrgdrng 24183 . . 3 ((Scalar‘𝑊) ∈ TopDRing → (Scalar‘𝑊) ∈ DivRing)
53, 4syl 17 . 2 (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ DivRing)
62islvec 21104 . 2 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing))
71, 5, 6sylanbrc 583 1 (𝑊 ∈ TopVec → 𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cfv 6560  Scalarcsca 17301  DivRingcdr 20730  LModclmod 20859  LVecclvec 21102  TopDRingctdrg 24166  TopVecctvc 24168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-lvec 21103  df-tdrg 24170  df-tlm 24171  df-tvc 24172
This theorem is referenced by: (None)
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