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

Theorem iscbn 30117
Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 24855 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 6892 . . . 4 (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2 iscbn.8 . . . 4 𝐷 = (IndMet‘𝑈)
31, 2eqtr4di 2791 . . 3 (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4 fveq2 6892 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5 iscbn.x . . . . 5 𝑋 = (BaseSet‘𝑈)
64, 5eqtr4di 2791 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
76fveq2d 6896 . . 3 (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
83, 7eleq12d 2828 . 2 (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9 df-cbn 30116 . 2 CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
108, 9elrab2 3687 1 (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  cfv 6544  CMetccmet 24771  NrmCVeccnv 29837  BaseSetcba 29839  IndMetcims 29844  CBanccbn 30115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-cbn 30116
This theorem is referenced by:  cbncms  30118  bnnv  30119  bnsscmcl  30121  cnbn  30122  hhhl  30457  hhssbnOLD  30532
  Copyright terms: Public domain W3C validator