Metamath Proof Explorer

Theorem rlmvsca

Description: Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015)

Ref Expression
Assertion rlmvsca 
|- ( .r ` R ) = ( .s ` ( 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 sravsca 
 |-  ( T. -> ( .r ` R ) = ( .s ` ( ringLMod ` R ) ) )
5 4 mptru 
 |-  ( .r ` R ) = ( .s ` ( ringLMod ` R ) )