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Metamath Proof Explorer

Theorem cvsclm

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

		Ref	Expression
	Hypothesis	cvslvec.1	$⊢ φ \to W \in ℂVec$
	Assertion	cvsclm	$⊢ φ \to W \in CMod$

Step	Hyp	Ref	Expression
1		cvslvec.1	$⊢ φ \to W \in ℂVec$
2		df-cvs	$⊢ ℂVec = CMod \cap LVec$
3	2	elin2	$⊢ W \in ℂVec \leftrightarrow (W \in CMod \land W \in LVec)$
4	3	simplbi	$⊢ W \in ℂVec \to W \in CMod$
5	1 4	syl	$⊢ φ \to W \in CMod$