Theorem hlnvi 25484

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 25483	. 2 |- ( U e. CHilOLD -> U e. NrmCVec )
3	1, 2	ax-mp 5	1 |- U e. NrmCVec

Syntax hints:   e. wcel 1767   NrmCVeccnv 25153   CHilOLDchlo 25477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-cbn 25455  df-hlo 25478

This theorem is referenced by:  htthlem  25510  axhfvadd-zf  25575  axhvcom-zf  25576  axhvass-zf  25577  axhvaddid-zf  25579  axhfvmul-zf  25580  axhvmulid-zf  25581  axhvmulass-zf  25582  axhvdistr1-zf  25583  axhvdistr2-zf  25584  axhvmul0-zf  25585  axhis2-zf  25588  axhis3-zf  25589  axhcompl-zf  25591  hilcompl  25794