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Theorem iscbn 28657
 Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 23952 instead. (New usage is discouraged.)
Hypotheses
Ref Expression
iscbn.x 𝑋 = (BaseSet‘𝑈)
iscbn.8 𝐷 = (IndMet‘𝑈)
Assertion
Ref Expression
iscbn (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Proof of Theorem iscbn
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6646 . . . 4 (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2 iscbn.8 . . . 4 𝐷 = (IndMet‘𝑈)
31, 2eqtr4di 2851 . . 3 (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4 fveq2 6646 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5 iscbn.x . . . . 5 𝑋 = (BaseSet‘𝑈)
64, 5eqtr4di 2851 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
76fveq2d 6650 . . 3 (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
83, 7eleq12d 2884 . 2 (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9 df-cbn 28656 . 2 CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
108, 9elrab2 3631 1 (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6325  CMetccmet 23868  NrmCVeccnv 28377  BaseSetcba 28379  IndMetcims 28384  CBanccbn 28655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-cbn 28656 This theorem is referenced by:  cbncms  28658  bnnv  28659  bnsscmcl  28661  cnbn  28662  hhhl  28997  hhssbnOLD  29072
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