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Theorem iscbn 21443
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 5525 . . . 4  |-  ( u  =  U  ->  ( IndMet `
 u )  =  ( IndMet `  U )
)
2 iscbn.8 . . . 4  |-  D  =  ( IndMet `  U )
31, 2syl6eqr 2333 . . 3  |-  ( u  =  U  ->  ( IndMet `
 u )  =  D )
4 fveq2 5525 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
5 iscbn.x . . . . 5  |-  X  =  ( BaseSet `  U )
64, 5syl6eqr 2333 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
76fveq2d 5529 . . 3  |-  ( u  =  U  ->  ( CMet `  ( BaseSet `  u
) )  =  (
CMet `  X )
)
83, 7eleq12d 2351 . 2  |-  ( u  =  U  ->  (
( IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) )  <-> 
D  e.  ( CMet `  X ) ) )
9 df-cbn 21442 . 2  |-  CBan  =  { u  e.  NrmCVec  |  (
IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) ) }
108, 9elrab2 2925 1  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255   CMetcms 18680   NrmCVeccnv 21140   BaseSetcba 21142   IndMetcims 21147   CBanccbn 21441
This theorem is referenced by:  cbncms  21444  bnnv  21445  bnsscmcl  21447  cnbn  21448  hhhl  21783  hhssbn  21857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-cbn 21442
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