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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 ∈ 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 )