Theorem hlnvi 30821

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 30820	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   ∈ wcel 2109  NrmCVeccnv 30513  CHil_OLDchlo 30814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-cbn 30792  df-hlo 30815

This theorem is referenced by:  htthlem  30846  axhfvadd-zf  30911  axhvcom-zf  30912  axhvass-zf  30913  axhvaddid-zf  30915  axhfvmul-zf  30916  axhvmulid-zf  30917  axhvmulass-zf  30918  axhvdistr1-zf  30919  axhvdistr2-zf  30920  axhvmul0-zf  30921  axhis2-zf  30924  axhis3-zf  30925  axhcompl-zf  30927  hilcompl  31130