MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tvctdrg Structured version   Visualization version   GIF version

Theorem tvctdrg 24201
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 24200 . 2 (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
32simprbi 496 1 (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  Scalarcsca 17300  TopDRingctdrg 24165  TopModctlm 24166  TopVecctvc 24167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-tvc 24171
This theorem is referenced by:  tvclvec  24207
  Copyright terms: Public domain W3C validator