Theorem hlnvi 27748

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 27747	. 2 |- ( U e. CHilOLD -> U e. NrmCVec )
3	1, 2	ax-mp 5	1 |- U e. NrmCVec

Syntax hints:   e. wcel 1990   NrmCVeccnv 27439   CHilOLDchlo 27741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-cbn 27719  df-hlo 27742

This theorem is referenced by:  htthlem  27774  axhfvadd-zf  27839  axhvcom-zf  27840  axhvass-zf  27841  axhvaddid-zf  27843  axhfvmul-zf  27844  axhvmulid-zf  27845  axhvmulass-zf  27846  axhvdistr1-zf  27847  axhvdistr2-zf  27848  axhvmul0-zf  27849  axhis2-zf  27852  axhis3-zf  27853  axhcompl-zf  27855  hilcompl  28058