Theorem tvctdrg 24474

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 24473	. 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
3	2	simprbi 503	1 ⊢ (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6527  Scalarcsca 17393  TopDRingctdrg 24438  TopModctlm 24439  TopVecctvc 24440

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-tvc 24444

This theorem is used by:  tvclvec  24480