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Metamath Proof Explorer

Theorem tvctlm

Description: A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015)

Ref Expression
Assertion tvctlm ( 𝑊 ∈ TopVec → 𝑊 ∈ TopMod )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
2 1 istvc ⊢ ( 𝑊 ∈ TopVec ↔ ( 𝑊 ∈ TopMod ∧ ( Scalar ‘ 𝑊 ) ∈ TopDRing ) )
3 2 simplbi ⊢ ( 𝑊 ∈ TopVec → 𝑊 ∈ TopMod )