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Theorem cbncms 26195
Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
iscbn.x  |-  X  =  ( BaseSet `  U )
iscbn.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
cbncms  |-  ( U  e.  CBan  ->  D  e.  ( CMet `  X
) )

Proof of Theorem cbncms
StepHypRef Expression
1 iscbn.x . . 3  |-  X  =  ( BaseSet `  U )
2 iscbn.8 . . 3  |-  D  =  ( IndMet `  U )
31, 2iscbn 26194 . 2  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )
43simprbi 462 1  |-  ( U  e.  CBan  ->  D  e.  ( CMet `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   ` cfv 5569   CMetcms 21985   NrmCVeccnv 25891   BaseSetcba 25893   IndMetcims 25898   CBanccbn 26192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-cbn 26193
This theorem is referenced by:  bnsscmcl  26198  ubthlem1  26200  ubthlem2  26201  minvecolem4a  26207  hlcmet  26224
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