Theorem hlnvi 26235

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 26234	. 2 |- ( U e. CHilOLD -> U e. NrmCVec )
3	1, 2	ax-mp 5	1 |- U e. NrmCVec

Syntax hints:   e. wcel 1844   NrmCVeccnv 25904   CHilOLDchlo 26228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-cbn 26206  df-hlo 26229

This theorem is referenced by:  htthlem  26261  axhfvadd-zf  26326  axhvcom-zf  26327  axhvass-zf  26328  axhvaddid-zf  26330  axhfvmul-zf  26331  axhvmulid-zf  26332  axhvmulass-zf  26333  axhvdistr1-zf  26334  axhvdistr2-zf  26335  axhvmul0-zf  26336  axhis2-zf  26339  axhis3-zf  26340  axhcompl-zf  26342  hilcompl  26545