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Theorem iscbn 25603
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 5872 . . . 4  |-  ( u  =  U  ->  ( IndMet `
 u )  =  ( IndMet `  U )
)
2 iscbn.8 . . . 4  |-  D  =  ( IndMet `  U )
31, 2syl6eqr 2526 . . 3  |-  ( u  =  U  ->  ( IndMet `
 u )  =  D )
4 fveq2 5872 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
5 iscbn.x . . . . 5  |-  X  =  ( BaseSet `  U )
64, 5syl6eqr 2526 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
76fveq2d 5876 . . 3  |-  ( u  =  U  ->  ( CMet `  ( BaseSet `  u
) )  =  (
CMet `  X )
)
83, 7eleq12d 2549 . 2  |-  ( u  =  U  ->  (
( IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) )  <-> 
D  e.  ( CMet `  X ) ) )
9 df-cbn 25602 . 2  |-  CBan  =  { u  e.  NrmCVec  |  (
IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) ) }
108, 9elrab2 3268 1  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5594   CMetcms 21561   NrmCVeccnv 25300   BaseSetcba 25302   IndMetcims 25307   CBanccbn 25601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-cbn 25602
This theorem is referenced by:  cbncms  25604  bnnv  25605  bnsscmcl  25607  cnbn  25608  hhhl  25944  hhssbn  26019
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