Theorem bnnv 31355

Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25574 instead. (New usage is discouraged.)

Assertion

Ref	Expression
bnnv	⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Proof of Theorem bnnv

Step	Hyp	Ref	Expression
1		eqid 2762	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2762	. . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
3	1, 2	iscbn 31353	. 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘(BaseSet‘𝑈))))
4	3	simplbi 502	1 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ‘cfv 6537  CMetccmet 25488  NrmCVeccnv 31073  BaseSetcba 31075  IndMetcims 31080  CBanccbn 31351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-cbn 31352

This theorem is used by:  bnrel  31356  bnsscmcl  31357  ubthlem1  31359  ubthlem2  31360  ubthlem3  31361  minvecolem1  31363  hlnv  31380