Metamath Proof Explorer

Theorem rlmds

Description: Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015)

Ref Expression
Assertion rlmds 
|- ( dist ` R ) = ( dist ` ( 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 srads 
 |-  ( T. -> ( dist ` R ) = ( dist ` ( ringLMod ` R ) ) )
5 4 mptru 
 |-  ( dist ` R ) = ( dist ` ( ringLMod ` R ) )