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Theorem hlnvi 21471
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 21470 . 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 1684   NrmCVeccnv 21140   CHil
OLDchlo 21464
This theorem is referenced by:  htthlem  21497  axhfvadd-zf  21562  axhvcom-zf  21563  axhvass-zf  21564  axhvaddid-zf  21566  axhfvmul-zf  21567  axhvmulid-zf  21568  axhvmulass-zf  21569  axhvdistr1-zf  21570  axhvdistr2-zf  21571  axhvmul0-zf  21572  axhis2-zf  21575  axhis3-zf  21576  axhcompl-zf  21578  hilcompl  21780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-cbn 21442  df-hlo 21465
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