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 ∈ DivRing → ( ringLMod ‘ 𝑅 ) ∈ LVec )

Proof

Step Hyp Ref Expression
1 drngring ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )
2 rlmlmod ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod )
3 1 2 syl ( 𝑅 ∈ DivRing → ( ringLMod ‘ 𝑅 ) ∈ LMod )
4 rlmsca ( 𝑅 ∈ DivRing → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) )
5 id ( 𝑅 ∈ DivRing → 𝑅 ∈ DivRing )
6 4 5 eqeltrrd ( 𝑅 ∈ DivRing → ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ∈ DivRing )
7 eqid ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) )
8 7 islvec ( ( ringLMod ‘ 𝑅 ) ∈ LVec ↔ ( ( ringLMod ‘ 𝑅 ) ∈ LMod ∧ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ∈ DivRing ) )
9 3 6 8 sylanbrc ( 𝑅 ∈ DivRing → ( ringLMod ‘ 𝑅 ) ∈ LVec )