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Theorem hlcmet 22388
Description: The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlcmet.x  |-  X  =  ( BaseSet `  U )
hlcmet.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
hlcmet  |-  ( U  e.  CHil OLD  ->  D  e.  ( CMet `  X
) )

Proof of Theorem hlcmet
StepHypRef Expression
1 hlobn 22382 . 2  |-  ( U  e.  CHil OLD  ->  U  e. 
CBan )
2 hlcmet.x . . 3  |-  X  =  ( BaseSet `  U )
3 hlcmet.8 . . 3  |-  D  =  ( IndMet `  U )
42, 3cbncms 22359 . 2  |-  ( U  e.  CBan  ->  D  e.  ( CMet `  X
) )
51, 4syl 16 1  |-  ( U  e.  CHil OLD  ->  D  e.  ( CMet `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446   CMetcms 19199   BaseSetcba 22057   IndMetcims 22062   CBanccbn 22356   CHil
OLDchlo 22379
This theorem is referenced by:  hlmet  22389  hlcompl  22409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-cbn 22357  df-hlo 22380
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