Theorem tvctlm

Description: A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Assertion	tvctlm	$$⊢W\in \mathrm{TopVec}\to W\in \mathrm{TopMod}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{Scalar}\left(W\right)=\mathrm{Scalar}\left(W\right)$$
2	1	istvc	$$⊢W\in \mathrm{TopVec}\leftrightarrow \left(W\in \mathrm{TopMod}\wedge \mathrm{Scalar}\left(W\right)\in \mathrm{TopDRing}\right)$$
3	2	simplbi	$$⊢W\in \mathrm{TopVec}\to W\in \mathrm{TopMod}$$