Theorem rlmsca

Description: Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014)

	Ref	Expression
Assertion	rlmsca	⊢ ( 𝑅 ∈ 𝑋 → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2	1	ressid	⊢ ( 𝑅 ∈ 𝑋 → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = 𝑅 )
3		rlmval	⊢ ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) )
4	3	a1i	⊢ ( 𝑅 ∈ 𝑋 → ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) ) )
5		ssidd	⊢ ( 𝑅 ∈ 𝑋 → ( Base ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) )
6	4 5	srasca	⊢ ( 𝑅 ∈ 𝑋 → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) )
7	2 6	eqtr3d	⊢ ( 𝑅 ∈ 𝑋 → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) )