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Theorem iscbn 30884
Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 25373 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 6905 . . . 4 (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2 iscbn.8 . . . 4 𝐷 = (IndMet‘𝑈)
31, 2eqtr4di 2794 . . 3 (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4 fveq2 6905 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5 iscbn.x . . . . 5 𝑋 = (BaseSet‘𝑈)
64, 5eqtr4di 2794 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
76fveq2d 6909 . . 3 (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
83, 7eleq12d 2834 . 2 (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9 df-cbn 30883 . 2 CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
108, 9elrab2 3694 1 (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  cfv 6560  CMetccmet 25289  NrmCVeccnv 30604  BaseSetcba 30606  IndMetcims 30611  CBanccbn 30882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-cbn 30883
This theorem is referenced by:  cbncms  30885  bnnv  30886  bnsscmcl  30888  cnbn  30889  hhhl  31224  hhssbnOLD  31299
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