Theorem bnnv 31196

Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25480 instead. (New usage is discouraged.)

Assertion

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

Proof of Theorem bnnv

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2763	. . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
3	1, 2	iscbn 31194	. 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘(BaseSet‘𝑈))))
4	3	simplbi 501	1 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   ∈ wcel 2143  ‘cfv 6538  CMetccmet 25394  NrmCVeccnv 30914  BaseSetcba 30916  IndMetcims 30921  CBanccbn 31192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-cbn 31193

This theorem is referenced by:  bnrel  31197  bnsscmcl  31198  ubthlem1  31200  ubthlem2  31201  ubthlem3  31202  minvecolem1  31204  hlnv  31221