Theorem tvctdrg 24226

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 24225	. 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
3	2	simprbi 500	1 ⊢ (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing)

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136  ‘cfv 6510  Scalarcsca 17265  TopDRingctdrg 24190  TopModctlm 24191  TopVecctvc 24192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-iota 6466  df-fv 6518  df-tvc 24196

This theorem is referenced by:  tvclvec  24232