Theorem bnnv 30814

Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25238 instead. (New usage is discouraged.)

Assertion

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

Proof of Theorem bnnv

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2729	. . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
3	1, 2	iscbn 30812	. 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘(BaseSet‘𝑈))))
4	3	simplbi 497	1 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   ∈ wcel 2109  ‘cfv 6482  CMetccmet 25152  NrmCVeccnv 30532  BaseSetcba 30534  IndMetcims 30539  CBanccbn 30810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-cbn 30811

This theorem is referenced by:  bnrel  30815  bnsscmcl  30816  ubthlem1  30818  ubthlem2  30819  ubthlem3  30820  minvecolem1  30822  hlnv  30839