Theorem cvsclm

Description: A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019)

	Ref	Expression
Hypothesis	cvslvec.1	$$⊢\phi \to W\in \mathrm{ℂVec}$$
Assertion	cvsclm	$$⊢\phi \to W\in \mathrm{CMod}$$

Proof

Step	Hyp	Ref	Expression
1		cvslvec.1	$$⊢\phi \to W\in \mathrm{ℂVec}$$
2		df-cvs	$$⊢\mathrm{ℂVec}=\mathrm{CMod}\cap \mathrm{LVec}$$
3	2	elin2	$$⊢W\in \mathrm{ℂVec}\leftrightarrow \left(W\in \mathrm{CMod}\wedge W\in \mathrm{LVec}\right)$$
4	3	simplbi	$$⊢W\in \mathrm{ℂVec}\to W\in \mathrm{CMod}$$
5	1 4	syl	$$⊢\phi \to W\in \mathrm{CMod}$$