Metamath Proof Explorer


Theorem nvctvc

Description: A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Assertion nvctvc ( 𝑊 ∈ NrmVec → 𝑊 ∈ TopVec )

Proof

Step Hyp Ref Expression
1 nvcnlm ( 𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod )
2 nlmtlm ( 𝑊 ∈ NrmMod → 𝑊 ∈ TopMod )
3 1 2 syl ( 𝑊 ∈ NrmVec → 𝑊 ∈ TopMod )
4 eqid ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
5 4 nlmnrg ( 𝑊 ∈ NrmMod → ( Scalar ‘ 𝑊 ) ∈ NrmRing )
6 1 5 syl ( 𝑊 ∈ NrmVec → ( Scalar ‘ 𝑊 ) ∈ NrmRing )
7 nvclvec ( 𝑊 ∈ NrmVec → 𝑊 ∈ LVec )
8 4 lvecdrng ( 𝑊 ∈ LVec → ( Scalar ‘ 𝑊 ) ∈ DivRing )
9 7 8 syl ( 𝑊 ∈ NrmVec → ( Scalar ‘ 𝑊 ) ∈ DivRing )
10 nrgtdrg ( ( ( Scalar ‘ 𝑊 ) ∈ NrmRing ∧ ( Scalar ‘ 𝑊 ) ∈ DivRing ) → ( Scalar ‘ 𝑊 ) ∈ TopDRing )
11 6 9 10 syl2anc ( 𝑊 ∈ NrmVec → ( Scalar ‘ 𝑊 ) ∈ TopDRing )
12 4 istvc ( 𝑊 ∈ TopVec ↔ ( 𝑊 ∈ TopMod ∧ ( Scalar ‘ 𝑊 ) ∈ TopDRing ) )
13 3 11 12 sylanbrc ( 𝑊 ∈ NrmVec → 𝑊 ∈ TopVec )