Theorem hlnvi 30911

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 30910	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   ∈ wcel 2108  NrmCVeccnv 30603  CHil_OLDchlo 30904

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-cbn 30882  df-hlo 30905

This theorem is referenced by:  htthlem  30936  axhfvadd-zf  31001  axhvcom-zf  31002  axhvass-zf  31003  axhvaddid-zf  31005  axhfvmul-zf  31006  axhvmulid-zf  31007  axhvmulass-zf  31008  axhvdistr1-zf  31009  axhvdistr2-zf  31010  axhvmul0-zf  31011  axhis2-zf  31014  axhis3-zf  31015  axhcompl-zf  31017  hilcompl  31220