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Theorem tvctdrg 24047
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
tvctdrg (π‘Š ∈ TopVec β†’ 𝐹 ∈ TopDRing)

Proof of Theorem tvctdrg
StepHypRef Expression
1 tlmtrg.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
21istvc 24046 . 2 (π‘Š ∈ TopVec ↔ (π‘Š ∈ TopMod ∧ 𝐹 ∈ TopDRing))
32simprbi 496 1 (π‘Š ∈ TopVec β†’ 𝐹 ∈ TopDRing)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6536  Scalarcsca 17206  TopDRingctdrg 24011  TopModctlm 24012  TopVecctvc 24013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-tvc 24017
This theorem is referenced by:  tvclvec  24053
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