Theorem bnnv 31255

Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25536 instead. (New usage is discouraged.)

Assertion

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

Proof of Theorem bnnv

Step	Hyp	Ref	Expression
1		eqid 2766	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2766	. . 3 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
3	1, 2	iscbn 31253	. 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 2146  ‘cfv 6543  CMetccmet 25450  NrmCVeccnv 30973  BaseSetcba 30975  IndMetcims 30980  CBanccbn 31251

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 2148  ax-9 2156  ax-ext 2738

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 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-cbn 31252

This theorem is used by:  bnrel  31256  bnsscmcl  31257  ubthlem1  31259  ubthlem2  31260  ubthlem3  31261  minvecolem1  31263  hlnv  31280