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Metamath Proof Explorer
Description: The scalar field of a topological vector space is a topological division
ring. (Contributed by Mario Carneiro, 5-Oct-2015)
|
|
Ref |
Expression |
|
Hypothesis |
tlmtrg.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
|
Assertion |
tvctdrg |
⊢ ( 𝑊 ∈ TopVec → 𝐹 ∈ TopDRing ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
tlmtrg.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
| 2 |
1
|
istvc |
⊢ ( 𝑊 ∈ TopVec ↔ ( 𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing ) ) |
| 3 |
2
|
simprbi |
⊢ ( 𝑊 ∈ TopVec → 𝐹 ∈ TopDRing ) |