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Theorem tvctdrg 23696
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 23695 . 2 (π‘Š ∈ TopVec ↔ (π‘Š ∈ TopMod ∧ 𝐹 ∈ TopDRing))
32simprbi 497 1 (π‘Š ∈ TopVec β†’ 𝐹 ∈ TopDRing)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  β€˜cfv 6543  Scalarcsca 17199  TopDRingctdrg 23660  TopModctlm 23661  TopVecctvc 23662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  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 6495  df-fv 6551  df-tvc 23666
This theorem is referenced by:  tvclvec  23702
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