Theorem hlnvi 31487

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 31486	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

This proof depends on syntax axioms:   ∈ wcel 2145  NrmCVeccnv 31179  CHil_OLDchlo 31480

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 2733

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 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  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 6493  df-fv 6545  df-cbn 31458  df-hlo 31481

This theorem is used by:  htthlem  31512  axhfvadd-zf  31577  axhvcom-zf  31578  axhvass-zf  31579  axhvaddid-zf  31581  axhfvmul-zf  31582  axhvmulid-zf  31583  axhvmulass-zf  31584  axhvdistr1-zf  31585  axhvdistr2-zf  31586  axhvmul0-zf  31587  axhis2-zf  31590  axhis3-zf  31591  axhcompl-zf  31593  hilcompl  31796