Theorem hlnvi 31373

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 31372	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

This proof depends on syntax axioms:   ∈ wcel 2145  NrmCVeccnv 31065  CHil_OLDchlo 31366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-cbn 31344  df-hlo 31367

This theorem is used by:  htthlem  31398  axhfvadd-zf  31463  axhvcom-zf  31464  axhvass-zf  31465  axhvaddid-zf  31467  axhfvmul-zf  31468  axhvmulid-zf  31469  axhvmulass-zf  31470  axhvdistr1-zf  31471  axhvdistr2-zf  31472  axhvmul0-zf  31473  axhis2-zf  31476  axhis3-zf  31477  axhcompl-zf  31479  hilcompl  31682