Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007) Use bnnvc instead. (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | bnnv | ⊢ ( 𝑈 ∈ CBan → 𝑈 ∈ NrmCVec ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid | ⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 ) | |
2 | eqid | ⊢ ( IndMet ‘ 𝑈 ) = ( IndMet ‘ 𝑈 ) | |
3 | 1 2 | iscbn | ⊢ ( 𝑈 ∈ CBan ↔ ( 𝑈 ∈ NrmCVec ∧ ( IndMet ‘ 𝑈 ) ∈ ( CMet ‘ ( BaseSet ‘ 𝑈 ) ) ) ) |
4 | 3 | simplbi | ⊢ ( 𝑈 ∈ CBan → 𝑈 ∈ NrmCVec ) |