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Theorem hlnvi 27132
 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 ∈ CHilOLD
Assertion
Ref Expression
hlnvi 𝑈 ∈ NrmCVec

Proof of Theorem hlnvi
StepHypRef Expression
1 hlnvi.1 . 2 𝑈 ∈ CHilOLD
2 hlnv 27131 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  NrmCVeccnv 26823  CHilOLDchlo 27125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-cbn 27103  df-hlo 27126 This theorem is referenced by:  htthlem  27158  axhfvadd-zf  27223  axhvcom-zf  27224  axhvass-zf  27225  axhvaddid-zf  27227  axhfvmul-zf  27228  axhvmulid-zf  27229  axhvmulass-zf  27230  axhvdistr1-zf  27231  axhvdistr2-zf  27232  axhvmul0-zf  27233  axhis2-zf  27236  axhis3-zf  27237  axhcompl-zf  27239  hilcompl  27442
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