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

Theorem iscbn 24265
Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
iscbn.x  |-  X  =  ( BaseSet `  U )
iscbn.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
iscbn  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )

Proof of Theorem iscbn
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 fveq2 5691 . . . 4  |-  ( u  =  U  ->  ( IndMet `
 u )  =  ( IndMet `  U )
)
2 iscbn.8 . . . 4  |-  D  =  ( IndMet `  U )
31, 2syl6eqr 2493 . . 3  |-  ( u  =  U  ->  ( IndMet `
 u )  =  D )
4 fveq2 5691 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
5 iscbn.x . . . . 5  |-  X  =  ( BaseSet `  U )
64, 5syl6eqr 2493 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
76fveq2d 5695 . . 3  |-  ( u  =  U  ->  ( CMet `  ( BaseSet `  u
) )  =  (
CMet `  X )
)
83, 7eleq12d 2511 . 2  |-  ( u  =  U  ->  (
( IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) )  <-> 
D  e.  ( CMet `  X ) ) )
9 df-cbn 24264 . 2  |-  CBan  =  { u  e.  NrmCVec  |  (
IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) ) }
108, 9elrab2 3119 1  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5418   CMetcms 20765   NrmCVeccnv 23962   BaseSetcba 23964   IndMetcims 23969   CBanccbn 24263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-cbn 24264
This theorem is referenced by:  cbncms  24266  bnnv  24267  bnsscmcl  24269  cnbn  24270  hhhl  24606  hhssbn  24681
  Copyright terms: Public domain W3C validator