Theorem hlnvi 26600

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. CHilOLD

Assertion

Ref	Expression
hlnvi	|- U e. NrmCVec

Proof of Theorem hlnvi

Step	Hyp	Ref	Expression
1		hlnvi.1	. 2 |- U e. CHilOLD
2		hlnv 26599	. 2 |- ( U e. CHilOLD -> U e. NrmCVec )
3	1, 2	ax-mp 5	1 |- U e. NrmCVec

Syntax hints:   e. wcel 1898   NrmCVeccnv 26259   CHilOLDchlo 26593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-iota 5569  df-fv 5613  df-cbn 26561  df-hlo 26594

This theorem is referenced by:  htthlem  26626  axhfvadd-zf  26691  axhvcom-zf  26692  axhvass-zf  26693  axhvaddid-zf  26695  axhfvmul-zf  26696  axhvmulid-zf  26697  axhvmulass-zf  26698  axhvdistr1-zf  26699  axhvdistr2-zf  26700  axhvmul0-zf  26701  axhis2-zf  26704  axhis3-zf  26705  axhcompl-zf  26707  hilcompl  26910