Theorem bnnv 30853

Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25273 instead. (New usage is discouraged.)

Assertion

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

Proof of Theorem bnnv

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2731	. . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
3	1, 2	iscbn 30851	. 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘(BaseSet‘𝑈))))
4	3	simplbi 497	1 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   ∈ wcel 2111  ‘cfv 6487  CMetccmet 25187  NrmCVeccnv 30571  BaseSetcba 30573  IndMetcims 30578  CBanccbn 30849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6443  df-fv 6495  df-cbn 30850

This theorem is referenced by:  bnrel  30854  bnsscmcl  30855  ubthlem1  30857  ubthlem2  30858  ubthlem3  30859  minvecolem1  30861  hlnv  30878