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	|- ( T. -> ( ringLMod ` R ) = ( ( subringAlg ` R ) ` ( Base ` R ) ) )
3		ssidd	|- ( T. -> ( Base ` R ) C_ ( Base ` R ) )
4	2 3	sramulr	|- ( T. -> ( .r ` R ) = ( .r ` ( ringLMod ` R ) ) )
5	4	mptru	|- ( .r ` R ) = ( .r ` ( ringLMod ` R ) )