Theorem tvctdrg

Description: The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypothesis	tlmtrg.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
Assertion	tvctdrg	⊢ ( 𝑊 ∈ TopVec → 𝐹 ∈ TopDRing )

Proof

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