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Metamath Proof Explorer

Theorem rlmlvec

Description: The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Assertion rlmlvec 
|- ( R e. DivRing -> ( ringLMod ` R ) e. LVec )

Proof

Step Hyp Ref Expression
1 drngring 
 |-  ( R e. DivRing -> R e. Ring )
2 rlmlmod 
 |-  ( R e. Ring -> ( ringLMod ` R ) e. LMod )
3 1 2 syl 
 |-  ( R e. DivRing -> ( ringLMod ` R ) e. LMod )
4 rlmsca 
 |-  ( R e. DivRing -> R = ( Scalar ` ( ringLMod ` R ) ) )
5 id 
 |-  ( R e. DivRing -> R e. DivRing )
6 4 5 eqeltrrd 
 |-  ( R e. DivRing -> ( Scalar ` ( ringLMod ` R ) ) e. DivRing )
7 eqid 
 |-  ( Scalar ` ( ringLMod ` R ) ) = ( Scalar ` ( ringLMod ` R ) )
8 7 islvec 
 |-  ( ( ringLMod ` R ) e. LVec <-> ( ( ringLMod ` R ) e. LMod /\ ( Scalar ` ( ringLMod ` R ) ) e. DivRing ) )
9 3 6 8 sylanbrc 
 |-  ( R e. DivRing -> ( ringLMod ` R ) e. LVec )