Theorem hlnvi 30828

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 30827	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   ∈ wcel 2109  NrmCVeccnv 30520  CHil_OLDchlo 30821

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-cbn 30799  df-hlo 30822

This theorem is referenced by:  htthlem  30853  axhfvadd-zf  30918  axhvcom-zf  30919  axhvass-zf  30920  axhvaddid-zf  30922  axhfvmul-zf  30923  axhvmulid-zf  30924  axhvmulass-zf  30925  axhvdistr1-zf  30926  axhvdistr2-zf  30927  axhvmul0-zf  30928  axhis2-zf  30931  axhis3-zf  30932  axhcompl-zf  30934  hilcompl  31137