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

Theorem iscbn 28945
Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 24235 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 6717 . . . 4 (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2 iscbn.8 . . . 4 𝐷 = (IndMet‘𝑈)
31, 2eqtr4di 2796 . . 3 (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4 fveq2 6717 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5 iscbn.x . . . . 5 𝑋 = (BaseSet‘𝑈)
64, 5eqtr4di 2796 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
76fveq2d 6721 . . 3 (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
83, 7eleq12d 2832 . 2 (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9 df-cbn 28944 . 2 CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
108, 9elrab2 3605 1 (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2110  cfv 6380  CMetccmet 24151  NrmCVeccnv 28665  BaseSetcba 28667  IndMetcims 28672  CBanccbn 28943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-cbn 28944
This theorem is referenced by:  cbncms  28946  bnnv  28947  bnsscmcl  28949  cnbn  28950  hhhl  29285  hhssbnOLD  29360
  Copyright terms: Public domain W3C validator