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Theorem iscbn 22371
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 5731 . . . 4  |-  ( u  =  U  ->  ( IndMet `
 u )  =  ( IndMet `  U )
)
2 iscbn.8 . . . 4  |-  D  =  ( IndMet `  U )
31, 2syl6eqr 2488 . . 3  |-  ( u  =  U  ->  ( IndMet `
 u )  =  D )
4 fveq2 5731 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
5 iscbn.x . . . . 5  |-  X  =  ( BaseSet `  U )
64, 5syl6eqr 2488 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
76fveq2d 5735 . . 3  |-  ( u  =  U  ->  ( CMet `  ( BaseSet `  u
) )  =  (
CMet `  X )
)
83, 7eleq12d 2506 . 2  |-  ( u  =  U  ->  (
( IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) )  <-> 
D  e.  ( CMet `  X ) ) )
9 df-cbn 22370 . 2  |-  CBan  =  { u  e.  NrmCVec  |  (
IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) ) }
108, 9elrab2 3096 1  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   ` cfv 5457   CMetcms 19212   NrmCVeccnv 22068   BaseSetcba 22070   IndMetcims 22075   CBanccbn 22369
This theorem is referenced by:  cbncms  22372  bnnv  22373  bnsscmcl  22375  cnbn  22376  hhhl  22711  hhssbn  22785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-cbn 22370
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