Theorem hlnvi 31501

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

Step	Hyp	Ref	Expression
1		hlnvi.1	. 2 ⊢ 𝑈 ∈ CHilOLD
2		hlnv 31500	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

This proof depends on syntax axioms:   ∈ wcel 2144  NrmCVeccnv 31193  CHil_OLDchlo 31494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6496  df-fv 6549  df-cbn 31472  df-hlo 31495

This theorem is used by:  htthlem  31526  axhfvadd-zf  31591  axhvcom-zf  31592  axhvass-zf  31593  axhvaddid-zf  31595  axhfvmul-zf  31596  axhvmulid-zf  31597  axhvmulass-zf  31598  axhvdistr1-zf  31599  axhvdistr2-zf  31600  axhvmul0-zf  31601  axhis2-zf  31604  axhis3-zf  31605  axhcompl-zf  31607  hilcompl  31810