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Theorem hlnvi 21487
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 21486 . 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 1696   NrmCVeccnv 21156   CHil
OLDchlo 21480
This theorem is referenced by:  htthlem  21513  axhfvadd-zf  21578  axhvcom-zf  21579  axhvass-zf  21580  axhvaddid-zf  21582  axhfvmul-zf  21583  axhvmulid-zf  21584  axhvmulass-zf  21585  axhvdistr1-zf  21586  axhvdistr2-zf  21587  axhvmul0-zf  21588  axhis2-zf  21591  axhis3-zf  21592  axhcompl-zf  21594  hilcompl  21796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-cbn 21458  df-hlo 21481
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