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Theorem iscbn 28328
 Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 23628 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 6545 . . . 4 (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2 iscbn.8 . . . 4 𝐷 = (IndMet‘𝑈)
31, 2syl6eqr 2851 . . 3 (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4 fveq2 6545 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5 iscbn.x . . . . 5 𝑋 = (BaseSet‘𝑈)
64, 5syl6eqr 2851 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
76fveq2d 6549 . . 3 (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
83, 7eleq12d 2879 . 2 (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9 df-cbn 28327 . 2 CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
108, 9elrab2 3624 1 (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  ‘cfv 6232  CMetccmet 23544  NrmCVeccnv 28048  BaseSetcba 28050  IndMetcims 28055  CBanccbn 28326 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-iota 6196  df-fv 6240  df-cbn 28327 This theorem is referenced by:  cbncms  28329  bnnv  28330  bnsscmcl  28332  cnbn  28333  hhhl  28668  hhssbnOLD  28743
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