MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bnnv Structured version   Visualization version   GIF version

Theorem bnnv 30895
Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25388 instead. (New usage is discouraged.)
Assertion
Ref Expression
bnnv (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Proof of Theorem bnnv
StepHypRef Expression
1 eqid 2735 . . 3 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 eqid 2735 . . 3 (IndMet‘𝑈) = (IndMet‘𝑈)
31, 2iscbn 30893 . 2 (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘(BaseSet‘𝑈))))
43simplbi 497 1 (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cfv 6563  CMetccmet 25302  NrmCVeccnv 30613  BaseSetcba 30615  IndMetcims 30620  CBanccbn 30891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-cbn 30892
This theorem is referenced by:  bnrel  30896  bnsscmcl  30897  ubthlem1  30899  ubthlem2  30900  ubthlem3  30901  minvecolem1  30903  hlnv  30920
  Copyright terms: Public domain W3C validator