Theorem lvecring

Description: The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023)

	Ref	Expression
Hypothesis	lvecring.1	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
Assertion	lvecring	⊢ ( 𝑊 ∈ LVec → 𝐹 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		lvecring.1	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		lveclmod	⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod )
3	1	lmodring	⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )
4	2 3	syl	⊢ ( 𝑊 ∈ LVec → 𝐹 ∈ Ring )