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Theorem tvctdrg 23689
Description: The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
tlmtrg.f 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
tvctdrg (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing)

Proof of Theorem tvctdrg
StepHypRef Expression
1 tlmtrg.f . . 3 𝐹 = (Scalar‘𝑊)
21istvc 23688 . 2 (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
32simprbi 498 1 (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6541  Scalarcsca 17197  TopDRingctdrg 23653  TopModctlm 23654  TopVecctvc 23655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6493  df-fv 6549  df-tvc 23659
This theorem is referenced by:  tvclvec  23695
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