Theorem clmlmod

Description: A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Assertion	clmlmod	⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
2		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
3	1 2	isclm	⊢ ( 𝑊 ∈ ℂMod ↔ ( 𝑊 ∈ LMod ∧ ( Scalar ‘ 𝑊 ) = ( ℂfld ↾s ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( SubRing ‘ ℂfld ) ) )
4	3	simp1bi	⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )