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Theorem iscbn 21459
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 5541 . . . 4  |-  ( u  =  U  ->  ( IndMet `
 u )  =  ( IndMet `  U )
)
2 iscbn.8 . . . 4  |-  D  =  ( IndMet `  U )
31, 2syl6eqr 2346 . . 3  |-  ( u  =  U  ->  ( IndMet `
 u )  =  D )
4 fveq2 5541 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
5 iscbn.x . . . . 5  |-  X  =  ( BaseSet `  U )
64, 5syl6eqr 2346 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
76fveq2d 5545 . . 3  |-  ( u  =  U  ->  ( CMet `  ( BaseSet `  u
) )  =  (
CMet `  X )
)
83, 7eleq12d 2364 . 2  |-  ( u  =  U  ->  (
( IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) )  <-> 
D  e.  ( CMet `  X ) ) )
9 df-cbn 21458 . 2  |-  CBan  =  { u  e.  NrmCVec  |  (
IndMet `  u )  e.  ( CMet `  ( BaseSet
`  u ) ) }
108, 9elrab2 2938 1  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271   CMetcms 18696   NrmCVeccnv 21156   BaseSetcba 21158   IndMetcims 21163   CBanccbn 21457
This theorem is referenced by:  cbncms  21460  bnnv  21461  bnsscmcl  21463  cnbn  21464  hhhl  21799  hhssbn  21873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-cbn 21458
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