Theorem tvctdrg 24311

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

Step	Hyp	Ref	Expression
1		tlmtrg.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	istvc 24310	. 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
3	2	simprbi 502	1 ⊢ (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ‘cfv 6525  Scalarcsca 17303  TopDRingctdrg 24275  TopModctlm 24276  TopVecctvc 24277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-tvc 24281

This theorem is referenced by:  tvclvec  24317