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 ) |