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Theorem hlcms 19322
Description: Every complex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
Assertion
Ref Expression
hlcms  |-  ( W  e.  CHil  ->  W  e. CMetSp
)

Proof of Theorem hlcms
StepHypRef Expression
1 hlbn 19319 . 2  |-  ( W  e.  CHil  ->  W  e. Ban )
2 bncms 19299 . 2  |-  ( W  e. Ban  ->  W  e. CMetSp )
31, 2syl 16 1  |-  ( W  e.  CHil  ->  W  e. CMetSp
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726  CMetSpccms 19287  Bancbn 19288   CHilchl 19289
This theorem is referenced by:  pjthlem2  19341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4215  df-iota 5420  df-fv 5464  df-bn 19291  df-hl 19292
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