Theorem rlmmulr

Description: Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015)

	Ref	Expression
Assertion	rlmmulr	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ ( ringLMod ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		rlmval	⊢ ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) )
2	1	a1i	⊢ ( ⊤ → ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) ) )
3		ssidd	⊢ ( ⊤ → ( Base ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) )
4	2 3	sramulr	⊢ ( ⊤ → ( .r ‘ 𝑅 ) = ( .r ‘ ( ringLMod ‘ 𝑅 ) ) )
5	4	mptru	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ ( ringLMod ‘ 𝑅 ) )