Theorem assasca

Description: The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015) (Revised by SN, 2-Mar-2025)

	Ref	Expression
Hypothesis	assasca.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
Assertion	assasca	⊢ ( 𝑊 ∈ AssAlg → 𝐹 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		assasca.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		assalmod	⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ LMod )
3	1	lmodring	⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )
4	2 3	syl	⊢ ( 𝑊 ∈ AssAlg → 𝐹 ∈ Ring )