Theorem rlmsca2

Description: Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015)

	Ref	Expression
Assertion	rlmsca2	⊢ ( I ‘ 𝑅 ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		fvi	⊢ ( 𝑅 ∈ V → ( I ‘ 𝑅 ) = 𝑅 )
2		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
3	2	ressid	⊢ ( 𝑅 ∈ V → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = 𝑅 )
4	1 3	eqtr4d	⊢ ( 𝑅 ∈ V → ( I ‘ 𝑅 ) = ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) )
5		fvprc	⊢ ( ¬ 𝑅 ∈ V → ( I ‘ 𝑅 ) = ∅ )
6		reldmress	⊢ Rel dom ↾s
7	6	ovprc1	⊢ ( ¬ 𝑅 ∈ V → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = ∅ )
8	5 7	eqtr4d	⊢ ( ¬ 𝑅 ∈ V → ( I ‘ 𝑅 ) = ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) )
9	4 8	pm2.61i	⊢ ( I ‘ 𝑅 ) = ( 𝑅 ↾s ( Base ‘ 𝑅 ) )
10		rlmval	⊢ ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) )
11	10	a1i	⊢ ( ⊤ → ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) ) )
12		ssidd	⊢ ( ⊤ → ( Base ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) )
13	11 12	srasca	⊢ ( ⊤ → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) )
14	13	mptru	⊢ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) )
15	9 14	eqtri	⊢ ( I ‘ 𝑅 ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) )