Metamath Proof Explorer


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 ) )