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Theorem hlnvi 22235
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 22234 . 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 1717   NrmCVeccnv 21904   CHil
OLDchlo 22228
This theorem is referenced by:  htthlem  22261  axhfvadd-zf  22326  axhvcom-zf  22327  axhvass-zf  22328  axhvaddid-zf  22330  axhfvmul-zf  22331  axhvmulid-zf  22332  axhvmulass-zf  22333  axhvdistr1-zf  22334  axhvdistr2-zf  22335  axhvmul0-zf  22336  axhis2-zf  22339  axhis3-zf  22340  axhcompl-zf  22342  hilcompl  22544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-cbn 22206  df-hlo 22229
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