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Theorem hlnvi 22386
Description: Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
hlnvi.1  |-  U  e. 
CHil OLD
Assertion
Ref Expression
hlnvi  |-  U  e.  NrmCVec

Proof of Theorem hlnvi
StepHypRef Expression
1 hlnvi.1 . 2  |-  U  e. 
CHil OLD
2 hlnv 22385 . 2  |-  ( U  e.  CHil OLD  ->  U  e.  NrmCVec )
31, 2ax-mp 8 1  |-  U  e.  NrmCVec
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   NrmCVeccnv 22055   CHil
OLDchlo 22379
This theorem is referenced by:  htthlem  22412  axhfvadd-zf  22477  axhvcom-zf  22478  axhvass-zf  22479  axhvaddid-zf  22481  axhfvmul-zf  22482  axhvmulid-zf  22483  axhvmulass-zf  22484  axhvdistr1-zf  22485  axhvdistr2-zf  22486  axhvmul0-zf  22487  axhis2-zf  22490  axhis3-zf  22491  axhcompl-zf  22493  hilcompl  22695
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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